1. Introduce slack variables for each inequaility to make them equations and rewrite the objective function in the form ax by cz... + P = 0.

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Constance Booth
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1 3.4 Simplex Method If a linear programming problem has more than 2 variables, solving graphically is not the way to go. Instead, we ll use a more methodical, numeric process called the Simplex Method. In this class, we ll only do the simplex method with what we call standard maximization problems. A standard maximization problem is one in which the objective function is being maximized, one which includes the real world inequality constraints x 0, y 0, z 0,..., and in which all of the other constraint inequalities are of the form ax + by + cz +... D where D is some non-negative number. We ll walk through the simplex process with the following example. Maximize P = 10x + 20y Subject to: x + 3y 36 5x + y 40 x 0 y 0 Note: Once you recognize it as a standard maximization problem, we ll ignore the real-world inequalities during the simplex process, although we know our answer must make sense. Steps to the Simplex Method for a Standard Maximization Problem: 1. Introduce slack variables for each inequaility to make them equations and rewrite the objective function in the form ax by cz... + P = Set up the simplex tableau (table) with the rewritten objective function in the bottom row below the horizontal line. Any variable corresponding to a unit column is called a basic variable. The other columns are called non-basic variables. A unit column is one that consists of ones and zeroes only. 1

2 3. Choose the pivot element. (a) Find the MOST NEGATIVE element in the last row. This determines the pivot column. If there are two that have the same most negative value, either one is fine. (b) For each POSITIVE entry above the horizontal line in the pivot column, divide the constant on the right side of the vertical line by the corresponding element in the pivot column. The row with the smallest ratio becomes the pivot row. (c) The element where the pivot column and row intersect is the pivot element Use the following website to type in your simplex tableau and pivot about the pivot element. (It is linked on my webpage.) Continue finding pivot elements and pivoting until THERE ARE NO NEGATIVE ENTRIES IN THE ROW BELOW THE HORIZONTAL LINE To find the solution, set all non-basic variables equal to 0 and all basic variables equal to the corresponding entry in the last column. 2

3 To see what this method is really doing, let s look at the tableaus and read off where we are at each step in the process by setting all non-basic variables equal to 0 and all basic variables equal to the corresponding entry in the last column /3 1 1/ /3 0 1/ /3 0 20/ /14 1/ /14 3/ /7 5/ We did the problem below graphically in the previous section and found there were leftover blocks of wood at the optimal solution. Maximize P = 10x + 16y Subject to: 2x + 2y 16 4x + 2y 20 x 2y y 7 x 0, y 0 3

4 x y s 1 s 2 s 3 s 4 P

5 Solve the linear programming problem using the simplex method. Maximize P = 10x + 9y + 5z Subject to: 3x + 2y 2z 3 4y + z 12 6x 5y + z 8 x 0, y 0, z 0 5

6 Solve the following linear programming problem using the simplex method. A firm manufactures three products: tables, desks, and bookcases. To produce each table requires 1 hour of labor, 10 square feet of wood, and 2 quarts of finish. To produce each desk requires 3 hours of labor, 35 square feet of wood, and 1 quart of finish. To produce each bookcase requires 0.50 hours of labor, 15 square feet of wood, and 1 quart of finish. Available is at most 25 hours of labor, at most 350 square feet of wood, and at most 55 quarts of finish. The table yields a profit of $4, the desk $3, and the bookcase $3. Find the number of each product to be made in order to maximize profits. What is the maximum profit? Are there any leftovers? 6

c) Place the Coefficients from all Equations into a Simplex Tableau, labeled above with variables indicating their respective columns

c) Place the Coefficients from all Equations into a Simplex Tableau, labeled above with variables indicating their respective columns BUILDING A SIMPLEX TABLEAU AND PROPER PIVOT SELECTION Maximize : 15x + 25y + 18 z s. t. 2x+ 3y+ 4z 60 4x+ 4y+ 2z 100 8x+ 5y 80 x 0, y 0, z 0 a) Build Equations out of each of the constraints above by introducing