Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e Copyright ©2015 Pearson Education, Inc. All.

Graphing Linear Inequalities in Two Variables
Section 7.1 Graphing Linear Inequalities in Two Variables Copyright ©2015 Pearson Education, Inc. All right reserved.

Linear Programming: The Graphical Method
Section 7.2 Linear Programming: The Graphical Method Copyright ©2015 Pearson Education, Inc. All right reserved.

Solve the following linear programming problem:
Example: Minimize subject to Solution: The feasible region is shown to the right. From the figure, the corner points are These corner points give the following values of z. The minimum value of z is 2; it occurs at Copyright ©2015 Pearson Education, Inc. All right reserved.

Applications of Linear Programming
Section 7.3 Applications of Linear Programming Copyright ©2015 Pearson Education, Inc. All right reserved.

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, Copyright ©2015 Pearson Education, Inc. All right reserved.

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 Copyright ©2015 Pearson Education, Inc. All right reserved.

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. Copyright ©2015 Pearson Education, Inc. All right reserved.

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. Copyright ©2015 Pearson Education, Inc. All right reserved.

The Simplex Method: Maximization
Section 7.4 The Simplex Method: Maximization Copyright ©2015 Pearson Education, Inc. All right reserved.

Determine the pivot in the simplex tableau given below.
Example: Determine the pivot in the simplex tableau given below. Solution: Look at the indicators (the last row of the tableau) and choose the most negative one. 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. Copyright ©2015 Pearson Education, Inc. All right reserved.

Determine the pivot in the simplex tableau given below.
Example: Determine the pivot in the simplex tableau given below. Quotients Solution: The entry in the pivot row and pivot column is the pivot. 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. In this case, the pivot is 2. Copyright ©2015 Pearson Education, Inc. All right reserved.

Maximization Applications
Section 7.5 Maximization Applications Copyright ©2015 Pearson Education, Inc. All right reserved.

Example: 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: Copyright ©2015 Pearson Education, Inc. All right reserved.

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. Copyright ©2015 Pearson Education, Inc. All right reserved.

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: Write the rows of matrix A as the columns of the transpose. Copyright ©2015 Pearson Education, Inc. All right reserved.

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 Copyright ©2015 Pearson Education, Inc. All right reserved.

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): Copyright ©2015 Pearson Education, Inc. All right reserved.

Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e Copyright ©2015 Pearson Education, Inc. All.

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Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e Copyright ©2015 Pearson Education, Inc. All.

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Presentation on theme: "Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e Copyright ©2015 Pearson Education, Inc. All."— Presentation transcript:

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