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 LectureDateScribe 1Tue, 29 MarchLewis Nerenberg 2Thu, 31 MarchDavid Bernick 3Tue, 5 AprilNikhil Bobb 4Thu, 7 AprilJake Kendall 5Thu, 7 April 6-8pmShaomin Ding 6Tue, 12 AprilNadya Sihaeva 7Thu, 14 AprilJun Liao 8Thu, 14 April 6-8pmZhenjiang Li -Week 19,21 April 9Tue, 26 April 10Thu, 28 AprilRolando Menchaca 11Thu, 28 April 6-8pmPritam Roy

Announcements LectureDateScribe 12Tue, 3 MayGeoff Ryder 13Thu, 5 MayNoah Wilson 14Thu, 5 May 6-8pmZuobing Xu 15Tue, 10 MayAdam Ryska 16Thu, 12 MayMarcelo Carvalho -Week of 17, 19 May 17Tue, 24 MayRyan Crabb 18Thu, 26 May 19Tue, 31 May 20Thu, 2 June 21Tue, 7 June 4:00 – 7:00 pm

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 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

Questions and Break

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

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

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)

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

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 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

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)