10/2 The simplex algorithm. In an augmented matrix, if a column has a 1 and all other entries 0, it is said to be ‘in solution’. The 1 is called a ‘pivot’ and the associated variable is a ‘basic’ variable The pivot operation is a combination of row operations that brings a column ‘into solution’. You can pivot on any nonzero entry. Example: Perform a pivot on the first row first column entry x y s t rhs The pivot can be used to go from one solution to a system to another solution.

Suppose we want to find the solution to the system with augmented matrix which maximizes P = 2x + y and keeps x,y,s and t non-negative. x y s t rhs Right now the basic variables are s and t. The nonbasic are x and y and the solution is (0,0,10,20) with P=0.

Using the simplex algorithm to solve linear optimization problems. First example: a problem that can be worked graphically. (See lecture 9 for the graphical solution.) Maximize: P = 5x + 8y + 6 Subject to: x,y>= 0 (1) x + 4y <=4 and (2) y + 4x <= 4. Introduce slack variables s1 >= 0 and s2 >= 0 to turn inequalities (1) and (2) in equations (1) and (2) (1) x + 4y + s1 = 4 and (2) y + 4x + s2 = 4 Now the problem is Find x,y,s1,s2>=0 satisfying Equations (1) and (2) for which P is as large As possible.

Tableau for the problem P=6 at (x,y,s1,s2)=(0,0,4,4) x y s1 s2 P rhs

Here is a problem that is harder to work graphically. Autoparts Inc produces 3 types of parts (L, M, N). Each type requires (2,5,3) units work on machine I And (2,2,5) units of work on machine II. The machines have Respectively (190, 150) units of work available. The profit From each part is (8,5,11) dollars respectively. Find a production schedule (x, y, z) that maximizes profits. Set up: Profit function Machine I constraint: Machine II constraint: Assume that the production of L must be greater than or Equal to the total production of M and N. The constraint is

Question: How do you spot an unbounded problem when you don’t have a picture?

Setting up linear programming problems: A farmer has 150 acres of land suitable for crops A and B. The cost of growing A is $40/acre. The cost for B is $60/acre The farmer has $7600 captial available. Each acre of A takes 20 hrs of labor and each acre of B takes 25 hrs of labor. The farmer has 3300 hrs of labor available. He expects to Make $150/acre for A and $200/acre for B. How many acres of each crop should he plant to maximize his profit?