10/2 The simplex algorithm.

In an augmented matrix, if a column has a 1 and all otherentries 0, it is said to be ‘in solution’. The 1 is called a

‘pivot’ and the associated variable is a ‘basic’ variableThe pivot operation is a combination of row operations thatbrings 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 2 5 1 0 10 1 2 0 1 20 The pivot can be used to go from one solution to a systemto another solution.

Suppose we want to find the solution to the system withaugmented matrix which maximizes P = 2x + yand keeps x,y,s and t non-negative.

x y s t rhs 2 5 1 0 10 1 2 0 1 20

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 turninequalities (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 satisfyingEquations (1) and (2) for which P is as largeAs possible.

Tableau for the problemP=6 at (x,y,s1,s2)=(0,0,4,4) x y s1 s2 P rhs1 4 1 0 0 44 1 0 1 0 4-5 8 0 0 1 6

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 IAnd (2,2,5) units of work on machine II. The machines haveRespectively (190, 150) units of work available. The profitFrom 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 orEqual to the total production of M and N. The constraint is

Question: How do you spot an unbounded problem whenyou 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/acreThe farmer has $7600 captial available. Each acre of A takes20 hrs of labor and each acre of B takes 25 hrs of labor.The farmer has 3300 hrs of labor available. He expects toMake $150/acre for A and $200/acre for B. How many acresof each crop should he plant to maximize his profit?