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3.4 Linear Programming

3.4 Linear Programming. p. 112. Learning Target. I can solve linear programing problem. Optimization -. Finding the minimum or maximum value of some quantity.

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3.4 Linear Programming

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  1. 3.4 Linear Programming p. 112

  2. Learning Target • I can solve linear programing problem.

  3. Optimization - • Finding the minimum or maximum value of some quantity. • Linear programming is a form of optimization where you optimize an objective function with a system of linear inequalities called constraints. • The overlapped shaded region is called the feasible region. • Infeasible – when the constraints of a linear programing application do not overlap. • Alternate optimal solutions- when there are two or more possible linear programing application (usually the graph is parallel to one side

  4. Solving a linear programming problem 1. Graph the constraints. 2. Locate the ordered pairs of the vertices of the feasible region. 3. If the feasible region is bounded (or closed), it will have a minimum & a maximum. If the region is unbounded (or open), it will have only one (a minimum OR a maximum). 4. Plug the vertices into the linear equation (C=) to find the min. and/or max.

  5. A note about: Unbounded Feasible Regions • If the region is unbounded, but has a top on it, there will be a maximum only. • If the region is unbounded, but has a bottom, there will be a minimum only.

  6. Vertices of feasible region: (2,8) C= -2+3(8)= 22 (2,0) C= -2+3(0)= -2 (5,0) C= -5+3(0)= -5 (5,2) C= -5+3(2)= 1 Find the min. & max. values of C=-x+3y subject to the following constraints. x  2 x  5 y  0 y  -2x+12 Max. of 22 at (2,8) Min. of -5 at (5,0)

  7. Ex: C=x+5y Find the max. & min. subject to the following constraints x0 y2x+2 5x+y • Vertices? (0,2) C=0+5(2)=10 (1,4) C=1+5(4)=21 Maximum only! Max of 21 at (1,4)

  8. Here is a plan of the steps used to solve word problems using linear programming: • After reading the question, make a chart to see the information more clearly. • Assign variables to the unknowns. • Form expressions to represent the restrictions. • Graph the inequalities. • Find the coordinates of the corner points of the feasible region. • Find the vertex point that maximizes or minimizes what we are looking for. • State the solution in a sentence. Theory – Solving Problems Using Linear Programming

  9. Example – Seven Steps

  10. Example – Seven Steps cont’d

  11. Example – Seven Steps cont’d

  12. Example 2 – Seven Steps

  13. Example 2 – Seven Steps cont’d

  14. Example 2 – Seven Steps cont’d

  15. Assignment Groupwork Every table will answer one problem on page 116 to 118. Remember to assign an equipment manager, recorder, presentor and facilitator. Write legibly your answer on a huge graphing paper. Equiptment manager get the materilas (3 colors of markers, Huge graphing paper and ruler.)

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