Decision Making via Linear Programming:A simple introduction

Fred Phillipsfred.phillips@stonybrook.edu

The nature of simple LP• One, single decision criterion

– Either maximize or minimize something– Usually, profit or cost

• No consideration of probability• Usually only one best answer

• Elaborations of LP allow for probabilities, multiple criteria, integer-only solutions, and more.– We’ll look at some of these elaborations later in the

semester.

Introducing LP via example

• Suppose we manufacture furniture.

• We must decide how many tables and how many chairs to make this week.

• We assume we can sell all that we make.– At a profit of $4/table and $3/chair.

Obviously we can’t make an infinite number of tables or chairs.

• Each table requires 4 hours processing in machine (or ‘department’) A...– ... and 2 hours in machine (dept.) B.

• Each chair requires 2 hours in dept. A...– ... and 4 hours in B.

• We have capacity constraints:– A total of 60 hours/week in dept. A, and– A total of 48 hours/week in dept. B.

Let’s write all this in algebraic form

• Let x1 be how many tables we make.• Let x2 be how many chairs we make.• We want to maximize

4x1 + 3x2

• Subject to our capacity constraints:4x1 + 2x2 < 60

2x1 + 4x2 < 48

• We also require that x1 > 0 and x2 > 0

Because your professor cleverly drew an example with only two

variables x1 and x2,• We can solve the

exercise using a 2D graph.

• This is not possible for realistic problems which may have several thousand variables.

The best (“optimal”) solution is always at a vertex!

(12,6) 4x+2y = 60; 2x+4y = 48 66 Maximum

(15,0) 4x+2y = 60; y = 0 60

(0,12) 2x+4y = 48; x = 0 36

(0,0) x = 0; y = 0 0

Vertex Lines Through Vertex Value of Objective

(Graph and solution generatedby http://www.zweigmedia.com/RealWorld/LPGrapher/lpg.html .The site uses x for tables and y for chairs.)

Another site will solve problems with many variables.• http://www.zweigmedia.com/RealWorld/simplex.html

Now let’s see how to solve these using Excel

• You need the “Solver” plug-in.

• Watch http://www.youtube.com/watch?v=0KPHmyyghew

Now you try it. Set up this model, and solve it in Excel.

A customer of the Regal Corporation needs 1,000 pounds of a chemical mixture consisting of three raw materials. Cost for each of these is as follows: • X1 = $2 per pound • X2 = $3 per pound • X3 = $4 per pound

The customer requires the mixture to meet these conditions: • The mix must contain at least 200 pounds of X2. • The mix cannot contain more than 400 pounds of X1. • The mix must contain at least 100 pounds of X3.

Determine the least-cost mixture for the batch of 1,000 pounds which will satisfy the customer's requirements.