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

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

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion Outline Solving Linear Programming Problems 1 Linear Programming Solution Procedure 2 Dealing with Unbounded Regions 3 Conclusion 4

Solving Linear Programming Problems Linear Programming Solution Procedure Dealing with Unbounded Regions Conclusion Outline Solving Linear Programming Problems 1 Linear Programming Solution Procedure 2 Dealing with Unbounded Regions 3 Conclusion 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 order to maximize profit. Objective: Maximize profit of 3 x + 7 y  10 x + 8 y 400 ≤   x + 3 y 100 ≤  Constraints: 0 x ≥   y 0  ≥

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 order to maximize profit. Objective: Maximize profit of 3 x + 7 y  10 x + 8 y 400 ≤   x + 3 y 100 ≤  Constraints: 0 x ≥   y 0  ≥

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

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

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

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