ISM 206 Lecture 2

1. ISM 206Lecture 2 Linear Programming and the Simplex Method

2. Announcements • Schedule for note taking is on web (may change) • First homework is on web

3. Announcements

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 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. Questions and Break

14. Solutions, Extreme points and bases • How many solutions are there to a set of linear equations? • Convexity of feasible region • Extreme points

15. 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

16. Solutions, Extreme points and bases • 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 • B gives us a basic solution • If this is feasible then it is called a basic feasible solution (bfs) or corner point feasible (cpf)

17. Optimality of a basis • B=feasible basis. A = [B, N] • Write LP in terms of basis

18. 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.

19. Simplex Method: basis change • One basic variable is replace by another • The optimality test identifies a non-basic variable to enter the basis • The entering variable is increased until one of the other basic variables becomes zero • This is found using the minimum ratio test • That variable departs the basis

20. Proof that the Simplex method works • If there exists an optimal point, there exists an optimal basic feasible solution • There are a finite number of bfs • Each iteration, the simplex method moves from one bfs to another, and always improves the objective function value • Therefore the simplex method must converge to the optimal solution (in at most S steps, where S is the number of basic feasible solutions)