1. The Simplex Method: Standard Minimization Problems
Section 4.2
The Simplex Method: Standard Minimization Problems

2. A Standard (minimization) Linear Programming Problem:
The objective function is to be minimized.
All the variables involved in the problem are nonnegative.
Each constraint may be written so that the expression with the variables is greater than or equal to a nonnegative constant.

3. The Dual Problem Theorem of Duality
Maximization problems can be associated with minimization problems (and vice versa). The original problem is called the Primal and the associated problem is called the Dual.
Theorem of Duality
A primal problem has a solution if and only if the dual has a solution.
Both objective functions attain the same optimal value.
The optimal solution of the primal appears under the slack variables in the last row of the final simplex tableau associated with the dual.

4. Ex.
Minimize C = 10x + 11y
Subject to
Write down the primal information:

5. Interchange the columns and rows and use variables u, v, w.
This can be represented by the problem:
Maximize P = 300u + 300v + 250w
Subject to
This is a standard maximization problem

6. Create the initial simplex tableau adding slack variables x and y.
pivot

9. This is the final tableau with x = 15, y = 5, and P = C = 205.