ISM 206 Lecture 2

1. ISM 206Lecture 2 Intro to Linear Programming

2. Announcements • Scribe Schedule on website

3. Next Four Lectures: Linear Programming • Properties of LP’s • The Simplex Method • Sensitivity and Duality • Alternative Methods for solving

4. Outline • Typical Linear Programming Problems • Standard Form • Converting Problems into standard form • Geometry of LP • Extreme points, linear independence and bases • Optimality Conditions • The simplex method • Graphically • Analytically

5. Product Mix Problem • How much beer and ale to produce from three scarce resources: • 480 pounds of corn • 160 ounces of hops • 1190 pounds of malt • A barrel of ale consumes 5 pounds of corn, 4 ounces of hops, 35 pounds of malt • A barrel of beer consumes 15 pounds of corn, 4 ounces of hops and 20 pounds of malt • Profits are $13 per barrel of ale, $23 for beer

6. Transportation Problem • A firm produces computers in Singapore and Hoboken. • Distribution Centers are in Oakland, Hong Kong and Istanbul • Supply, demand and costs summary:

7. Other LP examples • Blending problem • Diet problem • Assignment problem

8. Key Elements of LP’s • Proportionality • Additivity • Divisibility Building a Linear Model • Identify activities • Identify items • Identify input-output coefficients • Write the constraints • Identify coefficients of objective function

9. Geometry of LP • Consider the plot of solutions to a LP

10. Types of LP descriptions To deal with different types of objectives and constraints we convert each linear program to standard form.

11. Standard Form(according to Hillier and Lieberman) Concise version: A is an m by n matrix: n variables, m constraints

12. Converting into Standard Form • Slack/surplus variables • Replacing ‘free’ variables • Minimization vs maximization

13. Standard Form to Augmented Form A is an m by n matrix: n variables, m constraints

14. Questions and Break

15. Claim: We only need to worry about corner points (basic feasible solutions) • Proof: Assume there is a better interior point • This is a convex combination of 2 extreme points • Easy to show one must be at least as good

16. Basic Feasible Solutions • We have an equation Ax=b with more columns than rows • How do we normally solve this? • A basic solution corresponds to one that uses only linearly independent columns of A • A basic feasible solution is also feasible

17. Solutions, Extreme points and bases • Linear independence of vectors • Basis of a matrix • A basic solution of an LP • Basic Feasible solution (Corner Point Feasible): • The vector x is an extreme point of the solution space iff it is a bfs of Ax=b, x>=0 • Key fact: • If a LP has an optimal solution, then it has an optimal extreme point solution

18. Note on Rank of a matrix • Rank of a matrix = no. linearly independent cols (and rows) rank<=min{m,n} • A has full rank if rank(A)=m • If A is of full rank then there is at least one basis B of A • B is set of linearly independent columns of A • We will generally assume that A is of full rank

19. Simplex Method • Checks the corner points • Gets better solution at each iteration 1. Find a starting solution 2. Test for optimality • If optimal then stop 3. Perform one iteration to new CPF (BFS) solution. Back to 2.