Finite Mathematics MAT 141: Chapter 4 Notes Slack Variables and the - PowerPoint PPT Presentation

MAT 141, Chapter 4 Finite Mathematics MAT 141: Chapter 4 Notes Slack Variables and the Pivot The Simplex Method David J. Gisch S implex Method and S lack Variables S lack Variables • We use a slack variable to turn an inequality into an • We saw in the last chapter that we can use linear equality. programming to solve a practical problems. The only • For example let us look at issue is that the scope of linear programming is very � � � � � � 10 limited. • To tackle more complicated problems, and therefore • We change this equation into an equality by adding a more realistic, we need to introduce the simplex method nonnegative variable. and slack variables. � � � � � � � � 10 Here we know that � � � � � � 10 , so if � � � 3 and � � � 4 , then � � 3 . We call � a slack variable as it picks up the slack of the inequality to make an equality. 1

MAT 141, Chapter 4 S implex Method The Farmer Example: A farmer has 100 acres of land on which he wishes to plant a mixture of potatoes, corn, and cabbage. The amounts and constraints are given in the following table. • Because we are using several variables it is not (a) Write the objective function using subscripted convenient to use x, y, z etc. Thus, we use � � , read “x sub variables. one,” and so forth so we are not limited by the letters of � � 120� � � 40� � � 60� � the alphabet. The Farmer The Farmer Write the constraints as inequalities . (a) Write the constraints a s inequalities. a) (b) Basdkashflka (c) asjflhaljf b) Write the constraints as inequalities. (d) Now add the objective function with all variables � � � � � � � � � 100 moved to the left. 400� � � 160� � � 180� � � 20,000 � � �� � �� � �� � � 100 10� � � 4� � � 7� � � 500 10� � �4� � �7� � �� � � 500 Simplified �120� � �40� � �60� � �� � 0 c) Write each inequality as an equality using a different (e) Turn this into an augmented matrix. slack variable for each. The objective � � + � � + � � + � � � 100 � � � � � � � � � � � function 10� � + 4� � + 7� � + � � � 500 always goes 1 1 1 1 0 0 100 on the bottom. 10 4 7 0 1 0 500 �120 �40 �60 0 0 1 0 This augmented matrix is called the Sim plex Tableau . 2

MAT 141, Chapter 4 Bicycles Bicycles Example: A manufacturer of bicycles builds racing, touring, and mountain models. The bicycles are made of both aluminum and steel. The company has available 91,800 units of steel and 42,000 units of aluminum. The racing, touring, and mountain models need 17, 27, and 34 units of steel, and 12, 21, and 15 units of aluminum, respectively. How many of each type of bicycle should be made in order to maximize profit if the company makes $8 per racing bike, $12 per touring bike, and $22 per mountain bike? What is the maximum possible profit? (a) Write the objective function using subscripted variables. (b) Write the constraints as inequalities and turn them into equalities by including slack variables. (c) Take these equalities and the objective function and include them in a simplex tableau. The Pivot Method The Pivot Method • How do we take the simplex tableau and find a solution? • Read the solution from the result? � � � � � � � � � � � ▫ We use Gauss-Jordan to pivot about elements. 1 � 1 10 0 � � � ⁄ � � �� ⁄ 0 50 � 1 0 0 ▫ Pivot about the highlighted variable. 0 1 0 2 5 7 10 � �� 1 0 ⁄ 0 50 � � � � � � � � � � � � � 0 0 1 0 8 24 0 12 1 6000 1 1 1 1 0 0 100 � �� � � ` 10 4 7 0 1 0 500 • This tells us that with � � � 50 and � � � 50 we have a maximum �120 �40 �60 0 0 1 0 profit of $6000. • This means we should plant 50 acres of potatoes, no corn, and no � � � � � � � � � � � cabbage. �� � � � � 1 1 1 1 0 0 100 ▫ Thus, we end up leaving 50 acres unplanted (represented by the slack 2 5 7 10 1 0 � �� ⁄ 0 50 � � variable). It seems weird but it is actually optimal. Check it out. �120 �40 �60 0 0 1 0 120� � � � � Acres Cost Profit � � � � � � � � � � � 40 $16,000 $4,800 $4,000 Left � � � ⁄ � � �� ⁄ 1 � 1 10 0 0 50 � You have zeros above and 50 $20,000 $6,000 2 5 7 10 below. The pivot is 1 0 � �� ⁄ 0 50 � � 60 $24,000 $7,200 Out of Money complete. 0 8 24 0 12 1 6000 3

MAT 141, Chapter 4 The Pivot Method The Pivot Method Example: Pivot about the indicated number and state the Example: Pivot about the indicated number and state the resulting solution. resulting solution. � � � � � � � � � � � � � � � � � � � � � � � � � � 2 2 3 1 0 0 0 500 2 2 1 1 0 0 0 12 ` ` 4 1 1 0 1 0 0 300 1 2 3 0 1 0 0 45 7 2 4 0 0 1 0 700 3 1 1 0 0 1 0 20 �3 �4 �2 0 0 0 1 0 �2 �1 �3 0 0 0 1 0 S teps to the S implex Method Maximization Problems 4

MAT 141, Chapter 4 Find the Pivot Find the Pivot Example: Find the Pivot for the tableau. Example: Find the Pivot for the tableau. � � � � � � � � � � � � � � � � � � � � � � � � � � Smallest 12 1 22 3 2 2 1 1 0 0 0 12 4 2 3 1 0 0 0 22 � � 12 nonnegative � � 7.33 Smallest number. 1 2 3 0 1 0 0 45 45 3 2 2 5 0 1 0 0 28 28 5 � � 15 � � 5.6 nonnegative 3 1 1 0 0 1 0 20 20 1 1 3 2 0 0 1 0 45 45 2 � � 20 number. � � 22.5 �2 �1 �3 0 0 0 1 0 �3 �2 �4 0 0 0 1 0 Most negative indicator. Most negative indicator. Quick Review Find the Pivot Example: Find the Pivot for the tableau and perform the pivot. � � � � � � � � � � � � � 2 1 2 1 0 0 0 25 4 3 2 0 1 0 0 40 3 1 6 0 0 1 0 60 �4 �2 �3 0 0 0 1 0 4) 5

MAT 141, Chapter 4 Example Continued Example Continued Example Continued Put it all together. Example: Use the simplex method to solve the linear programming problem. Maximize: � � 8� � � 3� � � � � Subject to: � � � 6� � � 8� � � 118 � � � 5� � � 10� � � 220 � � � 0, � � � 0, � � � 0 6

MAT 141, Chapter 4 Example Continued Put it all together. � � � � � � � � � � � Example: Use the simplex method to solve the linear programming problem. 1 6 8 1 0 0 118 1 5 10 0 1 0 220 Maximize: � � 2� � � 5� � � � � �8 �3 �1 0 0 1 0 � � � 5� � � 2� � � 30 Subject to: 4� � � 3� � � 6� � � 72 � � � � � � � � � � � � � � 0, � � � 0, � � � 0 Remember to check 1 6 8 1 0 0 118 that all the �� � � � � → 0 �1 2 �1 1 0 2 indicators are 8� � � � � → 0 45 63 8 0 1 994 positive! Optimal solution is � � � 118 , � � � 0 , and � � � 0 which gives you a maximum of 994. Example Continued Put it all together. Example: Big Dave’s Fancy Widget Emporium makes three products. The distribution of labor and the profits are give in the table below. Create a simplex tableau and solve to maximize profit. Departm ent Production Hours by Product Departm ent Capacity for Hours A B C Assembling 2 3 2 30,000 Painting 1 2 2 38,000 Finishing 2 3 1 28,000 (PROFIT) $2 $5 $4 7

MAT 141, Chapter 4 Widgets Continued Toy Manufacturer Example: A small toy manufacturing firm has 200 squares of felt, 600 oz of stuffing, and 90 ft of trim available to make two types of toys, a small bear and a monkey. The bear requires 1 square of felt and 4 oz of stuffing. The monkey requires 2 squares of felt, 3 oz of stuffing, and 1 ft of trim. The firm makes $1 profit on each bear and $1.50 profit on each monkey. (a) Set up the linear programming problem to maximize profit. (b) Solve the linear programming problem. Toy Manufacturer Minimization Problems 8