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MATH 105: Finite Mathematics 3-2: Solving Linear Programming - - PowerPoint PPT Presentation

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Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion MATH 105: Finite Mathematics 3-2: Solving Linear Programming Problems Prof. Jonathan Duncan Walla Walla College Winter

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Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

MATH 105: Finite Mathematics 3-2: Solving Linear Programming Problems

  • Prof. Jonathan Duncan

Walla Walla College

Winter Quarter, 2006

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Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Outline

1

Solving Linear Programming Problems

2

Linear Programming Solution Procedure

3

Dealing with Unbounded Regions

4

Conclusion

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SLIDE 3

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Outline

1

Solving Linear Programming Problems

2

Linear Programming Solution Procedure

3

Dealing with Unbounded Regions

4

Conclusion

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SLIDE 4

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Solving a Linear Programming Problem

Recall that at the end of the last section, we found the corner points of the regions we graphed. Example Last time we set-up the following linear programming problem to determine how many batches of hot and mild salsa to make in

  • rder to maximize profit.

Objective: Maximize profit of 3x + 7y Constraints:        10x + 8y ≤ 400 x + 3y ≤ 100 x ≥ y ≥

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SLIDE 5

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Solving a Linear Programming Problem

Recall that at the end of the last section, we found the corner points of the regions we graphed. Example Last time we set-up the following linear programming problem to determine how many batches of hot and mild salsa to make in

  • rder to maximize profit.

Objective: Maximize profit of 3x + 7y Constraints:        10x + 8y ≤ 400 x + 3y ≤ 100 x ≥ y ≥

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SLIDE 6

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Finding the Optimal Solution

Example The graph for the critical region is shown below, along with the graph of 3x + 7y = C for several different Cs.

  • ✁
✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

As C changes, the line 3x + 7y = C moves. Unless it has the same slope as one of the boundary lines, it will first touch the region at a corner point.

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SLIDE 7

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Finding the Optimal Solution

Example The graph for the critical region is shown below, along with the graph of 3x + 7y = C for several different Cs.

  • ✁
✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

As C changes, the line 3x + 7y = C moves. Unless it has the same slope as one of the boundary lines, it will first touch the region at a corner point.

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SLIDE 8

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Finding the Optimal Solution

Example The graph for the critical region is shown below, along with the graph of 3x + 7y = C for several different Cs.

  • ✁
✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

As C changes, the line 3x + 7y = C moves. Unless it has the same slope as one of the boundary lines, it will first touch the region at a corner point.

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SLIDE 9

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Finding the Optimal Solution

Example The graph for the critical region is shown below, along with the graph of 3x + 7y = C for several different Cs.

  • ✁
✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

As C changes, the line 3x + 7y = C moves. Unless it has the same slope as one of the boundary lines, it will first touch the region at a corner point.

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SLIDE 10

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Finding the Optimal Solution

Example The graph for the critical region is shown below, along with the graph of 3x + 7y = C for several different Cs.

  • ✁
✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

As C changes, the line 3x + 7y = C moves. Unless it has the same slope as one of the boundary lines, it will first touch the region at a corner point.

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Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Completing The Example

Example Thus, to find the optimal number of batches of hot and mild salsa to make, we need only check the corner points to see which produces the most profit. Corner Point Profit

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Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Completing The Example

Example Thus, to find the optimal number of batches of hot and mild salsa to make, we need only check the corner points to see which produces the most profit. Corner Point Profit (0, 0) 3(0) + 7(0) = 0

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Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Completing The Example

Example Thus, to find the optimal number of batches of hot and mild salsa to make, we need only check the corner points to see which produces the most profit. Corner Point Profit (0, 0) 3(0) + 7(0) = 0 (10, 0) 3(0) + 7(10) = 70

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SLIDE 14

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Completing The Example

Example Thus, to find the optimal number of batches of hot and mild salsa to make, we need only check the corner points to see which produces the most profit. Corner Point Profit (0, 0) 3(0) + 7(0) = 0 (10, 0) 3(0) + 7(10) = 70

  • 0, 100

3

  • 3(0) + 7

100

3

  • = 233.3
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SLIDE 15

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Completing The Example

Example Thus, to find the optimal number of batches of hot and mild salsa to make, we need only check the corner points to see which produces the most profit. Corner Point Profit (0, 0) 3(0) + 7(0) = 0 (10, 0) 3(0) + 7(10) = 70

  • 0, 100

3

  • 3(0) + 7

100

3

  • = 233.3

200

11 , 300 11

  • 3

200

11

  • + 7

300

11

  • = 245.45
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Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Completing The Example

Example Thus, to find the optimal number of batches of hot and mild salsa to make, we need only check the corner points to see which produces the most profit. Corner Point Profit (0, 0) 3(0) + 7(0) = 0 (10, 0) 3(0) + 7(10) = 70

  • 0, 100

3

  • 3(0) + 7

100

3

  • = 233.3

200

11 , 300 11

  • 3

200

11

  • + 7

300

11

  • = 245.45

Note that because this is a story problem,

200 11 and 300 11 are not

possible answers. If you try (18, 27), which is close, you find a maximum profit of 243.

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SLIDE 17

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Outline

1

Solving Linear Programming Problems

2

Linear Programming Solution Procedure

3

Dealing with Unbounded Regions

4

Conclusion

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SLIDE 18

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Solving a Linear Programming Problem

From this example, we can develop a general procedure for solving a linear programming problem. Linear Programming Problems To solve a linear programming problem:

1 Graph the region given by the constraints. 2 Find the corner points of the region. 3 Evaluate the objective function at each corner point. 4 Pick the “best” corner point, if it exists.

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SLIDE 19

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Solving a Linear Programming Problem

From this example, we can develop a general procedure for solving a linear programming problem. Linear Programming Problems To solve a linear programming problem:

1 Graph the region given by the constraints. 2 Find the corner points of the region. 3 Evaluate the objective function at each corner point. 4 Pick the “best” corner point, if it exists.

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SLIDE 20

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Solving a Linear Programming Problem

From this example, we can develop a general procedure for solving a linear programming problem. Linear Programming Problems To solve a linear programming problem:

1 Graph the region given by the constraints. 2 Find the corner points of the region. 3 Evaluate the objective function at each corner point. 4 Pick the “best” corner point, if it exists.

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SLIDE 21

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Solving a Linear Programming Problem

From this example, we can develop a general procedure for solving a linear programming problem. Linear Programming Problems To solve a linear programming problem:

1 Graph the region given by the constraints. 2 Find the corner points of the region. 3 Evaluate the objective function at each corner point. 4 Pick the “best” corner point, if it exists.

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SLIDE 22

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Solving a Linear Programming Problem

From this example, we can develop a general procedure for solving a linear programming problem. Linear Programming Problems To solve a linear programming problem:

1 Graph the region given by the constraints. 2 Find the corner points of the region. 3 Evaluate the objective function at each corner point. 4 Pick the “best” corner point, if it exists.

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SLIDE 23

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Solving a Linear Programming Problem

From this example, we can develop a general procedure for solving a linear programming problem. Linear Programming Problems To solve a linear programming problem:

1 Graph the region given by the constraints. 2 Find the corner points of the region. 3 Evaluate the objective function at each corner point. 4 Pick the “best” corner point, if it exists.

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SLIDE 24

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

An Example with Multiple Solutions

Example Solve the following linear programming problem. Objective: Minimize 2x + 6y Constraints:        x + 3y ≤ 30 3x − 4y ≥ −27 x ≥ y ≥

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SLIDE 25

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Outline

1

Solving Linear Programming Problems

2

Linear Programming Solution Procedure

3

Dealing with Unbounded Regions

4

Conclusion

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Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

An Unbounded Region

Example Solve the following linear programming problem. Objective: Maximize 2x − y Constraints:        2x + 3y ≥ 6 3x − 2y ≥ −6 x ≥ 1 y ≥ 1

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Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Unbounded Regions

If the region formed by the constrains is unbounded, there is one more step to the solution process. Solution Procedure

1 Graph the feasible set given by the system of equations. 2 Find the corner points of the region (if empty, no solution). 3 Check the objective function at each corner point. 4 If the region is bounded, the solution is the corner point with

the min/max value. If two corner points have the same min/max value, there are infinitely many solutions.

5 If the region is unbounded, and your min/max value is found

  • n a line to ∞, pick an extra point farther along that line and

check it. If it is your new min/max point, then there is no

  • solution. Otherwise, the original corner point is your solution.
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SLIDE 28

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Unbounded Regions

If the region formed by the constrains is unbounded, there is one more step to the solution process. Solution Procedure

1 Graph the feasible set given by the system of equations. 2 Find the corner points of the region (if empty, no solution). 3 Check the objective function at each corner point. 4 If the region is bounded, the solution is the corner point with

the min/max value. If two corner points have the same min/max value, there are infinitely many solutions.

5 If the region is unbounded, and your min/max value is found

  • n a line to ∞, pick an extra point farther along that line and

check it. If it is your new min/max point, then there is no

  • solution. Otherwise, the original corner point is your solution.
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SLIDE 29

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Unbounded Regions

If the region formed by the constrains is unbounded, there is one more step to the solution process. Solution Procedure

1 Graph the feasible set given by the system of equations. 2 Find the corner points of the region (if empty, no solution). 3 Check the objective function at each corner point. 4 If the region is bounded, the solution is the corner point with

the min/max value. If two corner points have the same min/max value, there are infinitely many solutions.

5 If the region is unbounded, and your min/max value is found

  • n a line to ∞, pick an extra point farther along that line and

check it. If it is your new min/max point, then there is no

  • solution. Otherwise, the original corner point is your solution.
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SLIDE 30

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Unbounded Regions

If the region formed by the constrains is unbounded, there is one more step to the solution process. Solution Procedure

1 Graph the feasible set given by the system of equations. 2 Find the corner points of the region (if empty, no solution). 3 Check the objective function at each corner point. 4 If the region is bounded, the solution is the corner point with

the min/max value. If two corner points have the same min/max value, there are infinitely many solutions.

5 If the region is unbounded, and your min/max value is found

  • n a line to ∞, pick an extra point farther along that line and

check it. If it is your new min/max point, then there is no

  • solution. Otherwise, the original corner point is your solution.
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SLIDE 31

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Unbounded Regions

If the region formed by the constrains is unbounded, there is one more step to the solution process. Solution Procedure

1 Graph the feasible set given by the system of equations. 2 Find the corner points of the region (if empty, no solution). 3 Check the objective function at each corner point. 4 If the region is bounded, the solution is the corner point with

the min/max value. If two corner points have the same min/max value, there are infinitely many solutions.

5 If the region is unbounded, and your min/max value is found

  • n a line to ∞, pick an extra point farther along that line and

check it. If it is your new min/max point, then there is no

  • solution. Otherwise, the original corner point is your solution.
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SLIDE 32

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Unbounded Regions

If the region formed by the constrains is unbounded, there is one more step to the solution process. Solution Procedure

1 Graph the feasible set given by the system of equations. 2 Find the corner points of the region (if empty, no solution). 3 Check the objective function at each corner point. 4 If the region is bounded, the solution is the corner point with

the min/max value. If two corner points have the same min/max value, there are infinitely many solutions.

5 If the region is unbounded, and your min/max value is found

  • n a line to ∞, pick an extra point farther along that line and

check it. If it is your new min/max point, then there is no

  • solution. Otherwise, the original corner point is your solution.
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SLIDE 33

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Unbounded Regions

If the region formed by the constrains is unbounded, there is one more step to the solution process. Solution Procedure

1 Graph the feasible set given by the system of equations. 2 Find the corner points of the region (if empty, no solution). 3 Check the objective function at each corner point. 4 If the region is bounded, the solution is the corner point with

the min/max value. If two corner points have the same min/max value, there are infinitely many solutions.

5 If the region is unbounded, and your min/max value is found

  • n a line to ∞, pick an extra point farther along that line and

check it. If it is your new min/max point, then there is no

  • solution. Otherwise, the original corner point is your solution.
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SLIDE 34

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Outline

1

Solving Linear Programming Problems

2

Linear Programming Solution Procedure

3

Dealing with Unbounded Regions

4

Conclusion

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SLIDE 35

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Important Concepts

Things to Remember from Section 3-2

1 The solution process for bounded regions 2 The solution process for an unbounded region

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SLIDE 36

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Important Concepts

Things to Remember from Section 3-2

1 The solution process for bounded regions 2 The solution process for an unbounded region

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SLIDE 37

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Important Concepts

Things to Remember from Section 3-2

1 The solution process for bounded regions 2 The solution process for an unbounded region

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SLIDE 38

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Next Time. . .

Next time we will see our last section in this class. In that section, we will solve linear programming story problems using the methods seen today. For next time Read Section 3-3

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SLIDE 39

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion

Next Time. . .

Next time we will see our last section in this class. In that section, we will solve linear programming story problems using the methods seen today. For next time Read Section 3-3