1. Decision Making via Linear Programming: A simple introduction Fred Phillips fred.phillips@stonybrook.edu

2. 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.

3. 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.

4. 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.

5. Let’s write all this in algebraic form Let x 1 be how many tables we make. Let x 2 be how many chairs we make. We want to maximize 4x 1 + 3x 2 Subject to our capacity constraints: 4x 1 + 2x 2 < 60 2x 1 + 4x 2 < 48 We also require that x 1 > 0 and x 2 > 0

6. Because your professor cleverly drew an example with only two variables x 1 and x 2, We can solve the exercise using a 2D graph. This is not possible for realistic problems which may have several thousand variables.

7. The best (“optimal”) solution is always at a vertex! (12,6)4x+2y = 60; 2x+4y = 4866 Maximum (15,0)4x+2y = 60; y = 060 (0,12)2x+4y = 48; x = 036 (0,0)x = 0; y = 00 Vertex Lines Through Vertex Value of Objective (Graph and solution generated by http://www.zweigmedia.comhttp://www.zweigmedia.com /RealWorld/LPGrapher/lpg.html/RealWorld/LPGrapher/lpg.html. The site uses x for tables and y for chairs.)

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

9. Now let’s see how to solve these using Excel You need the “Solver” plug-in. Watch http://www.youtube.com/watch?v=0KPH myyghew http://www.youtube.com/watch?v=0KPH myyghew

10. 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.