Introduction to Linear Programming: Concepts, Methods, and Applications
This lecture covers the fundamentals of Linear Programming (LP), including the properties of LPs, the Simplex Method, and problem-solving techniques. Key topics include the conversion of problems to standard form, geometry of LP, optimality conditions, and various LP applications such as product mix, transportation, blending, diet, and assignment problems. The session emphasizes building linear models and understanding basic feasible solutions, corner points, and the role of linear independence in LP solutions.
Introduction to Linear Programming: Concepts, Methods, and Applications
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Presentation Transcript
ISM 206Lecture 2 Intro to Linear Programming
Announcements • Scribe Schedule on website
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:
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
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.
