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3.4 Linear Programming

Learn the concept of linear programming and how it can be used to optimize an objective function with a system of linear inequalities. Discover how to graph the constraints, locate the vertices of the feasible region, and find the minimum and maximum values. Apply this knowledge to solve real-life scenarios such as buying concert tickets.

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3.4 Linear Programming

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  1. 3.4 Linear Programming p. 163

  2. 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.

  3. Solving a linear programming problem • Graph the constraints. • Locate the ordered pairs of the vertices of the feasible region. • 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.

  4. 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. A note about: Unbounded Feasible Regions

  5. x  2 x  5 y  0 y  -2x+12 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. Max. of 22 at (2,8) Min. of -5 at (5,0)

  6. 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) Ex: C=x+5y Find the max. & min. subject to the following constraints

  7. Linear Programming • You want to buy tickets for some concerts; rock and jazz. • You want at least 5 rock tickets and at least 8 jazz tickets. • No more than 20 total tickets. • Rock tickets cost $3 and jazz tickets cost $5. • You cannot spend more than $80. • Find the number of each type of ticket you can buy at the lowest price total.

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