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3.4 Linear Programming Objective:
Find the maximum and minimum values of a function over a region.
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Linear Programming Linear programming involves using a system of inequalities to find the maximum and minimum values to solve real-world problems. Constraints – the actual lines of the inequalities Feasible Region – the overlapping region of solutions Vertices – the maximum or minimum of the feasible region. Bounded – when the region is enclosed (a polygon)
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Illustrations Constraints vertices Feasible Region
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Example Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function f(x, y)=3x-2y for this region. x≤5 y≤4 x+y≥2 or y≥-x+2
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Continued x≤5 y≤4 x+y≥2 or y≥-x+2 Find the vertices
y=-5+2 (substitute 5 for x) y=-3 (5, -3) 4=-x+2 (substitute 4 for y) x=-2 (-2, 4) The first two lines intersect at (5, 4) Use these vertices in the given function to find max and min. Maximum of 21 at (5,-3) Minimum of -14 at (-2,4) (x, y) 3x-2y f(x,y) (5, -3) 3(5)-2(-3) 21 (-2, 4) 3(-2)-2(4) -14 (5, 4) 3(5)-2(4) 7
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Try One y≥1 2≤x≤4 X-2y≥-4 f(x,y)=3y+x Solve for y (or x if there’s no y) 2≤x and x≤4 (change to x≥2) -2y≥-x-4 so y≤½x+2
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Find vertices (x, y) 3y+x f(x,y) (2,1) 2(1)+2 4 (4,1) 3(1)+4 7 (2,3)
Some are easy (2,1) (4,1) Solve to find exact for other two: y≤½x+2 intersects x≥2 y=½(2) + 2 y=3 intersects at (2,3) y≤½x+2 intersects x≤4 y=½(4)+2 intersects at (4, 4) Check the vertices with the given function Maximum: 16 at (4,4) Minimum: 4 at (2,1) (x, y) 3y+x f(x,y) (2,1) 2(1)+2 4 (4,1) 3(1)+4 7 (2,3) 3(3)+2 11 (4,4) 3(4)+4 16
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Unbounded x≥-3 y≤1 3x+y≤6 (y≤-3x+6) f(x, y)=5x-2y Graph first Region is unbounded Vertex: (-3, 1) Vertex: 1=-3x+6 -5=-3x X=5/3 (5/3, 1)
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Continued Check to see about maximums and minimums. To determine if these are maximum/minimum points, check other points in the region. Choose (0,0) to see what happens in the function, f(x,y)=5(0)-2(0)= 0 This is larger than the minimum. Try (2,-3) f(x,y)=5(2)-2(-3)=16. This is larger than the above “maximum” in the table. That means that there is no maximum for this unbounded region. The minimum is at (-3,1) (x,y) 5x-2y f(x,y) -3,1 5(-3)-2(1) -17 5/3, 1 5(5/3)-2(1) 19/3
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A landscaping company has crews who mow lawns and prune shrubbery
A landscaping company has crews who mow lawns and prune shrubbery. The company schedules 1 hour for mowing and hours for pruning jobs. Each crew is scheduled for no more than 2 pruning jobs per day. Each crew’s schedule is set up for a maximum of 9 hours per day. On the average, the charge for mowing a lawn is $40 and the charge for pruning a lawn is $120. Find a combination of mowing and pruning that will maximize the income the company receives per day for one of its crews.
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Homework: Page odd
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