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ISM 206 Lecture 2

ISM 206 Lecture 2. Linear Programming and the Simplex Method. Announcements. Schedule for note taking is on web (may change) First homework is on web. Announcements. Outline. Typical Linear Programming Problems Standard Form Converting Problems into standard form Geometry of LP

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ISM 206 Lecture 2

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  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)

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