Math 141 Lecture Notes

Section 4.1 The Simplex Method: Standard Maximization Problems

The Simplex Method

The Simplex Method is based on the Gauss-Jordan elimination method and provides solutions to programming problems when the number of variables or constraints is large. Beginning at some initial feasible solution, each iteration progresses to another corner point with an improved value of the objective function. The iteration is terminated when the optimal solution is reached, if it exists.

A standard maximization problem is one in which 1) The objective function is to be maximized. 2) All the variables involved in the problem are nonnegative. 3) Each linear constraint may be written so that the expression involving the variables is less than or equal to a nonnegative constant.

A slack variable is a nonnegative variable inserted in order to write a linear equation corresponding to a linear inequality to set up the Simplex Method.

The variables associated with unit columns are called basic variables; all other variables are called nonbasic variables.

A solution obtained by setting the nonbasic variables equal to zero is a basic solution of the system.

Selecting the Pivot Element 1) Select the pivot column: Locate the most negative entry to the left of the vertical line in the last row. The column containing this entry is the pivot column. If there is more than one such column, choose any one. 2) Select the pivot row: Divide each positive entry in the pivot column into its corresponding entry in the column of constants. The pivot row is the row corresponding to the smallest ratio thus obtained. If there is more than one such entry, choose any one. The pivot element is the element common to both the pivot column and the pivot row.

Example 1: Maximize P = 3x + 2y Subject to 2x + 3y ≤ 12 2x + y ≤ 8 x ≥ 0, y ≥ 0 Setting Up the Initial Simplex Tableau 1) Transform the system of linear inequalities into a system of linear equations by introducing slack variables. 2) Rewrite the objective function

P  c1 x1  c2 x2    cn xn in the form

 c1 x1  c2 x2    cn xn  P  0 where all the variables are on the left and the coefficient of P is +1. Write this equation below the equations of step 1. 3) Write the augmented matrix associated with this system of linear equations.

The Simplex Method 1) Set up the initial simplex tableau. 2) Determine whether the optimal solution has been reached by examining all entries in the last row to the left of the vertical line. a. If all the entries are nonnegative, the optimal solution has been reached. Proceed to step 4. b. If there are one or more negative entries, the optimal solution has not been reached. Proceed to step 3. 3) Perform the pivot operation. Locate the pivot element and convert it to a 1 by dividing all the elements in the pivot row by the pivot element. Using row operations, convert the pivot column into a unit column by adding suitable multiplies of the pivot row to each of the other rows as required. Return to step 2. 4) Determine the optimal solution(s). The value of the variable heading each unit column is given by the entry lying in the column of constants in the row containing the 1. The variables heading columns not in unit form are assigned the value zero. Example 2: Maximize: P = x + 1.2y Subject to 2x + y ≤ 180 x + 3y ≤ 300 x ≥ 0, y ≥ 0 Example 3: Minimize P = 2x + 2y + z Subject to 2x + y + 2z ≤ 14 2x + 4y + z ≤ 26 x + 2y + 3z ≤ 28 x ≥ 0, y ≥ 0, z ≥ 0 Example 4: Minimize P = 20x + 12y + 18z Subject to 3x + y + 2z ≤ 9 2x + 3y + z ≤ 8 x + 2y + 3z ≤ 7 x ≥ 0, y ≥ 0, z ≥ 0 Example 5: Ace Novelty Company has determined that the profits are $6, $5, and $4 for each type-A, type-B, and type-C souvenir that it plans to produce. To manufacture a type-A souvenir requires 2 minutes on machine I, 1 minute on machine II, and 2 minutes on machine III. A type-B souvenir requires 1 minute on machine I, 3 minutes on machine II, and 1 minute on machine III. A type-C souvenir requires 1 minute on machine I and 2 minutes on each of machines II and III. Each day there are 3 hours available on machine I, 5 hours available on machine II, and 4 hours available on machine III for manufacturing these souvenirs. How many souvenirs of each type should Ace Novelty make per day in order to maximize its profit?

Problems with Multiple Solutions and Problems with No Solutions

A linear programming problem may have infinitely many solutions iff the last row to the left of the vertical line of the final simplex tableau has a zero in a column that is not a unit column. A linear programming problem will have no solution if the simplex method breaks down at some stage. For example, if at some stage there are no nonnegative ratios in the computation, then the linear programming problem has no solution.