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3.4 Linear Programming p. 163

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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.

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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.

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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.

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Find the min. & max. values of C=-x+3y subject to the following constraints. 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 Max. of 22 at (2,8) Min. of -5 at (5,0)

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Ex: C=x+5y Find the max. & min. subject to the following constraints x0x0 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)

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Assignment

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