ISM 206 Lecture 3 The Simplex Method

Announcements Homework due 6pm Thursday Thursday 6pm lecture

Outline LP so far Correction: Standard form Why we can look only at basic feasible solutions Optimality conditions The simplex method The step from one bfs to the next Tableu method Phase I: Finding an initial BFS

LP so far Formulated LP’s in various contexts Transform any LP into a standard form LP Intuition of simplex method: Find the best corner point feasible solution Math required: –Corner point Feasible or basic feasible solutions correspond to a set of n active constraints –Any set of active constraints corresponds to a basis from the matrix A –The basis is a set of linearly independent columns

Standard Form Concise version: A is an m by n matrix: n variables, m constraints

Solutions, Extreme points and bases Key fact: –If a LP has an optimal solution, then it has an optimal extreme point solution (proved today) 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 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)

Optimal basis theorem Theorem If a LP in standard form has a finite optimal solution then it has an optimal basic feasible solution Proof Requires the representation theorem…

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.

Simplex Method: basis change One basic variable is replaced 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

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

The simplex method Example Table version

Questions and Break

A basic feasible solution B=basis of A. Write LP in terms of basis X is a basic solution of the LP X is a basic feasible solution if it is feasible! (example)

Optimality of a basis We want to test of a basic feasible solution is optimal Use the basic notation from before

Optimality test

Finding an initial bfs The ‘phase 1’ approach The ‘big M’ method

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)