Lecture2 - Linear Programs A linear function is a function...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
Linear Programs z A linear function is a function of the form: f(x 1 , x 2 , . . . , x n ) = c 1 x 1 + c 2 x 2 + . . . + c n x n = i=1 to n c i x i e.g., 3x 1 + 4x 2 -3x 4 . z A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities. e.g., 3x 1 + 4x 2 4 7 x 1 -2 x 5 = 7 z Typically, an LP has non-negativity constraints.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Terminology z Decision variables : e.g., x and y. In general, these are quantities you can control to improve your objective which should completely describe the set of decisions to be made. z Constraints : e.g., 2x + 3y 24 , x 0,y 0 Limitations on the values of the decision variables. z Objective Function . e.g., 4x + 5y Value measure used to rank alternatives Seek to maximize or minimize this objective examples: maximize NPV, minimize cost
Background image of page 2
Linear Programming Assumptions Maximize 4 K + 3 S 2 K + 3 S 10 …. Divisibility Assumption Each variable is allowed to assume fractional values. Certainty Assumption. Each linear coefficient of the objective function and constraints is known (and is not a random variable).
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Max’s linear program Step 2. Determine the objective function Step 3. Determine the constraints Maximize z = 4x + 5y (objective function) subject to 2x + 3y 60 (constraint) x 0 ; y 0 (non-negativity constraints)
Background image of page 4
23 Max’s linear program Step 2. Determine the objective function Step 3. Determine the constraints Maximize z = 4x + 5y (objective function) subject to 2x + 3y 60 (constraint) x 0 ; y 0 (non-negativity constraints) A feasible solution satisfies all of the constraints. x = 10, y = 10 is feasible; x = 10, y = 15 is infeasible . An optimal solution is the best feasible solution. The optimal solution is x = 30, y = 0.
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
MSR Marketing Inc. adapted from Frontline Systems Need to choose ads to reach at least 1.5 million people Minimize Cost Upper bound on number of ads of each type TV Radio Mail Newspaper Audience Size 50,000 25,000 20,000 15,000 Cost/Impression $500 $200 $250 $125 Max # of ads 20 15 10 15
Background image of page 6
Formulating as a math model 1. The decisions are how many ads of each type to choose. Let x 1 be the number of TV ads selected. Let x 2 , x 3 , x 4 denote the number of radio, mail, and newspaper ads. These are the “decision variables.” 2. What is the objective? Express the objective in terms of the decision variables.
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/23/2009 for the course ENGR 62 taught by Professor Unknown during the Spring '06 term at Stanford.

Page1 / 40

Lecture2 - Linear Programs A linear function is a function...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online