Copyright © 2019 Pearson Education, Inc. Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 12e Finite Mathematics with Applications 12e Copyright © 2019 Pearson Education, Inc. Slide 1 ALWAYS LEARNING

Chapter 7 Linear Programming

Graphing Linear Inequalities in Two Variables Section 7.1 Graphing Linear Inequalities in Two Variables

Example: Solution: Graph Figure 7.5 Solve the inequality for y: The graph consists of all points on or below the boundary line as shown below. Figure 7.5

Figure 7.15

Linear Programming: The Graphical Method Section 7.2 Linear Programming: The Graphical Method

Solve the following linear programming problem: Example: Minimize subject to Solution: First, we graph the feasible region is shown in Figure 7.21. To find the corner points, you must solve these three systems of equations:

Solve the following linear programming problem: Example: Minimize subject to Solving the two equations to find point A by substitution yields Similarly, solving the two equations to find Finding the last corner point C can also be found by substitution, and we can obtain

Solve the following linear programming problem: Example: Minimize subject to

Solve the following linear programming problem: Example: Minimize subject to These corner points give the value of z. The minimum of z is , it occurs at

Applications of Linear Programming Section 7.3 Applications of Linear Programming

Example: Certain laboratory animals must have at least 30 grams of protein and at least 20 grams of fat per feeding period. These nutrients come from food A, which costs 18¢ per unit and supplies 2 grams of protein and 4 of fat, and food B with 6 grams of protein and 2 of fat, costing 12¢ per unit. Food B is bought under a long-term contract requiring that at least 2 units of B be used per serving. How much of each food must be bought to produce the minimum cost per serving? Solution: Let x represent the amount of food A needed and y the amount of food B. Use the given information to produce the following table. Use the table to develop the linear programming problem. Since the animals must have at least 30 grams of protein and 20 grams of fat, use in the constraint inequalities for protein and fat. The long-term contract provides a constraint not in the table, namely,

Example: Certain laboratory animals must have at least 30 grams of protein and at least 20 grams of fat per feeding period. These nutrients come from food A, which costs 18¢ per unit and supplies 2 grams of protein and 4 of fat, and food B with 6 grams of protein and 2 of fat, costing 12¢ per unit. Food B is bought under a long-term contract requiring that at least 2 units of B be used per serving. How much of each food must be bought to produce the minimum cost per serving? Solution: So we have the following problem: Cost Protein Contract Fat (The constraint is redundant because of the constraint

Example: Certain laboratory animals must have at least 30 grams of protein and at least 20 grams of fat per feeding period. These nutrients come from food A, which costs 18¢ per unit and supplies 2 grams of protein and 4 of fat, and food B with 6 grams of protein and 2 of fat, costing 12¢ per unit. Food B is bought under a long-term contract requiring that at least 2 units of B be used per serving. How much of each food must be bought to produce the minimum cost per serving? Solution: A graph of the feasible region with the corner points identified is shown below. Figure 7.24

Example: Certain laboratory animals must have at least 30 grams of protein and at least 20 grams of fat per feeding period. These nutrients come from food A, which costs 18¢ per unit and supplies 2 grams of protein and 4 of fat, and food B with 6 grams of protein and 2 of fat, costing 12¢ per unit. Food B is bought under a long-term contract requiring that at least 2 units of B be used per serving. How much of each food must be bought to produce the minimum cost per serving? Solution: Use the corner point theorem to find the minimum value of z as shown in the table. The minimum value of 1.02 occurs at (3, 4). Thus, 3 units of food A and 4 units of food B will produce a minimum cost of $1.02 per serving.

The Simplex Method: Maximization Section 7.4 The Simplex Method: Maximization

Determine the pivot in the simplex tableau given below. Example: Determine the pivot in the simplex tableau given below. Solution: The most negative indicator identifies the variable that is to be eliminated from all but one of the equations (rows)—in this case, x2. The column containing the most negative indicator is called the pivot column.

Determine the pivot in the simplex tableau given below. Example: Determine the pivot in the simplex tableau given below. Solution: Now, for each positive entry in the pivot column, divide the number in the far right column of the same row by the positive number in the pivot column. The row with the smallest quotient (in this case, the second row) is called the pivot row. The entry in the pivot row and pivot column is the pivot. In this case, the pivot is 2.

Determine the pivot in the simplex tableau given below. Example: Determine the pivot in the simplex tableau given below. Solution: The row with the smallest quotient (in this case, the second row) is called the pivot row. The entry in the pivot row and pivot column is the pivot.

Continued on next slide

Continued from previous slide

Maximization Applications Section 7.5 Maximization Applications

Example: Solution: (a) What is the optimal solution? A chemical plant makes three products—glaze, solvent, and clay—each of which brings in different revenue per truckload. Production is limited, first by the number of air pollution units the plant is allowed to produce each day and second by the time available in the evaporation tank. The plant manager wants to maximize the daily revenue. Using information not given here, he sets up an initial simplex tableau and uses the simplex method to produce the following final simplex tableau: The three variables represent the number of truckloads of glaze, solvent, and clay, respectively. The first slack variable comes from the air pollution constraint and the second slack variable from the time constraint on the evaporation tank. The revenue function is given in hundreds of dollars. (a) What is the optimal solution? Solution:

Example: The plant manager produced the following final simplex tableau: The three variables represent the number of truckloads of glaze, solvent, and clay, respectively. The first slack variable comes from the air pollution constraint and the second slack variable comes from the time constraint on the evaporation tank. The revenue function is given in hundreds of dollars. (b) Interpret the solution. What do the variables represent, and what does the solution mean? Solution: The variable x1 is the number of truckloads of glaze, x2 the number of truckloads of solvent, x3 the number of truckloads of clay to be produced, and z the revenue produced (in hundreds of dollars). The plant should produce about 24 truckloads of clay and no glaze or solvent, for a maximum revenue of $9600. The first slack variable, s1, represents the number of air pollution units below the maximum number allowed. Since the number of air pollution units will be 60 less than the allowable maximum. The second slack variable, s2 , represents the unused time in the evaporation tank. Since the evaporation tank is fully used.

The Simplex Method: Duality and Minimization Section 7.6 The Simplex Method: Duality and Minimization

Find the transpose of each matrix. Example: Find the transpose of each matrix. (a) Solution: Write the rows of matrix A as the columns of the transpose. (b) Solution:

The Simplex Method: Nonstandard Problems Section 7.7 The Simplex Method: Nonstandard Problems

Example: Restate the following problem in terms of equations, and write its initial simplex tableau: Maximize subject to Solution: In order to write the constraints as equations, subtract a surplus variable from the constraint and add a slack variable to each constraint. So the problem becomes

Example: Restate the following problem in terms of equations, and write its initial simplex tableau: Maximize subject to Solution: Write the objective function as and use the coefficients of the four equations to write the initial simplex tableau (omitting the z column):