## The Simplex Wizard

**What it teaches:**

The Simplex Wizard app teaches the Simplex Method of solving linear programming problems. It shows
(1) how to *set up* the initial tableau for the problem and (2) how to *solve* the
problem using Gaussian elimination. The wizard allows you to focus on the method and
not get bogged down by the arithmetic. At every point in the procedure the status bar
at the bottom of the wizard offers context-sensitive help on what to do next.

**How it works:**

We will demonstrate using this linear programming problem: minimize the cost,

*C*= 5

*x*

_{1}+ 3

*x*

_{2},

subject to the constraints:

**Step 1. Set up the initial tableau.** The wizard asks you some general questions
such as whether the objective is to maximize or minimize, how many variables appear in the constraints
and how many and of what type are the constraints:

This is enough information for the wizard to create a tableau of the right size and to
partially fill it in:

You finish setting up the tableau by filling in the names of the variables and the
coefficients that appear in the constraints and objective function.

Now you can hover the mouse over each of the grids that make up the tableau
and the status bar shows how to interpret the contents of that grid.

**Step 2. Apply the Simplex Method.**
You choose an entering variable (column), you choose a leaving
variable (row), the wizard does Gaussian elimination. These three operations are repeated
until the solution is found. The status bar display helps you choose.

Here is the display at the point where you have to
choose an entering variable. Notice that the wizard says “choose x1”.

Now you have to pick an leaving variable (row). Notice that the wizard says “choose
a3”.

Now the wizard does Gaussian elimination on the tableau using the element at the
intersection of the entering column and leaving row as pivot.

This process – choosing an entering column, a leaving row,
Gaussian elimination – is repeated until the objective function is optimized
as shown here:

This final tableau says that the solution to our problem is a minimum cost of C = 23 and that this happens when x1 = 4 and x2 = 1.

Some other features of the Simplex Wizard are that you can save this simplex problem to a file, you can specify the number of significant figures to be displayed and you can print a report showing the steps that were used to solve the problem.

**Some ideas for teachers:**

- This application provides a good example of how systems of equations arise in economics.

- Using this app you can create more realistic problems that have more than just two or three constraints.

**Where to now?**