17_Review_of_LP

# 17_Review_of_LP - 15.082 and 6.855J Review of Linear...

This preview shows pages 1–12. Sign up to view the full content.

1 15.082 and 6.855J Review of Linear Programming

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 Overview Describe LP and IP min cost flow as an LP Graphical solution Basic feasible solutions. Simplex Method Basic feasible solutions in matrix form Duality Note: this will cover lots of material. We will also have a recitation.
3 The Minimum Cost Flow Problem u ij = capacity of arc (i,j). c ij = unit cost of shipping flow from node i to node j on (i,j). x ij = amount shipped on arc (i,j) Minimize (i,j) A c ij x ij j x ij - k x ki = b i for all i N . and 0 x ij u ij for all (i,j) A.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
4 Terminology j x ij - k x ki = b i for all i N . and 0 x ij u ij for all (i,j) A. x ij = Decision variable . Describes a decision to be made Minimize (i,j) A c ij x ij Objective Function Constraints
5 Terminology j x ij - k x ki = b i for all i N . and 0 x ij u ij for all (i,j) A. Minimize (i,j) A c ij x ij Objective Function Constraints In a linear program, the objective function and the constraints are all linear. Typically, but not always, the variables are constrained to be non-negative. If variables are constrained to be integers, it is called an integer program.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
6 Production Planning : Given several products with varying production requirements and cost structures, determine how much of each product to produce in order to maximize profits. Scheduling : Given a staff of people, determine an optimal work schedule that maximizes worker preferences while adhering to scheduling rules. Portfolio Management : Determine bond portfolios that maximize expected return subject to constraints on risk levels and diversification. And an incredible number more. Some Applications of LPs + IPs
7 Graphing 2-Dimensional LPs Example 1: x 3 0 1 2 y 0 1 2 4 3 Feasible Region x 0 y 0 x + 2 y 2 y 4 x 3 Subject to: Maximize x + y Optimal Solution These LP animations were created by Keely Crowston.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
8 Graphing 2-Dimensional LPs Example 2: Feasible Region x 0 y 0 -2 x + 2 y 4 x 3 Subject to: Minimize ** x - y Multiple Optimal Solutions! 4 1 x 3 1 2 y 0 2 0 3 1/3 x + y 4
9 Graphing 2-Dimensional LPs Example 3: Feasible Region x 0 y 0 x + y 20 x 5 -2 x + 5 y 150 Subject to: Minimize x + 1/3 y Optimal Solution x 30 10 20 y 0 10 20 40 0 30 40

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
y x 0 1 2 3 4 0 1 2 3 x 30 10 20 y 0 10 20 40 0 30 40 Do We Notice Anything From These 3 Examples? x y 0 1 2 3 4 0 1 2 3
A Fundamental Point If an optimal solution exists, there is always a corner point optimal solution! y

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 05/05/2010 for the course EE 15.082 taught by Professor Orlin during the Spring '10 term at Visayas State University.

### Page1 / 44

17_Review_of_LP - 15.082 and 6.855J Review of Linear...

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online