ISM 206 Lecture 2 Intro to Linear Programming

Announcements Scribe Schedule on website LectureDateTopicReadingScribeAssessment 1Thu, Sep 21Introduction and ModelingCh 1&2Damien Eads 2Tue, Sep 26Intro to Linear ProgrammingCh 3, 4, 5Alexe Bogdan 3Thu, Sep 27The simplex methodCh 6Shane BrennanHomework 1 assigned 4Tue, Oct 3Duality and Sensitivity AnalysisCh 7John Conners 5Wed, Oct 4 10am Other LP Methods. Transportation, Assignment and Network Optimization Problems Ch 8 &9Karen Glocer Thu, Oct 5No Class. Instructor away 6Tue, Oct 10Unconstrained Nonlinear Optimization Brett GyarfasHomework 1 due Homework 2 assigned 7Thu, Oct 12Nonlinear ProgrammingCh 12Prabath Gunawardane 8Tue, Oct 17Nonlinear Programming 2 Mike Schuresko 9Thu, Oct 19Nonlinear Programming 3 Hui ZhangHomework 2 due Homework 3 assigned

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

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

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

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

Other LP examples Blending problem Diet problem Assignment problem

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

Geometry of LP Consider the plot of solutions to a LP

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

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

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

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

Questions and Break

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

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

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

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

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.