LESSON 6-5: LINEAR OPTIMIZATION  Solve problems using linear programming.  Recognize situations in which there are no solutions or more than one solution of a linear programming application.

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optimization linear programming objective function constraints feasible solutions multiple optimal solutions unbounded

Find the maximum and minimum values of the objective function 𝒇(𝒙, 𝒚) and for what values of 𝒙 and 𝒚 they occur, subject to the given constraints.

f  x, y   5 x  2 y

x  3y  6 2x  y  8 x0 y0

Automotive: Mechanics at a repair garage carry two name brands of tires United and Royale. The number of Royale tires sold is typically less than or equal to twice the number of United tires sold. The shop can store at most 500 tires at one time. Due to factory capacity, the number of Royale tires produced is less than or equal to 50 more than 0.25 times the number of United tires. The garage earns $25 profit for each United tire and $20 profit for each Royale tire. A. Write an objective function and a list of constraints that model the given situation.

B. Sketch a graph of the region determined by the constraints to find how many of each tire the garage should sell to maintain optimal profit.

Find the maximum and minimum values of the objective function 𝒇(𝒙, 𝒚) and for what values of 𝒙 and 𝒚 they occur, subject to the given constraints.

f  x, y   4 x  8 y

x  2 y  16 y2 x3

Gorpian Manufacturing has two factories that produce three grades of paper: low grade, medium grade, and high grade. It needs to supply 24 tons of low grade, 6 tons of medium grade, and 30 tons of high grade paper. Factory A produces 8 tons of low grade, 1 ton of medium grade, and 2 tons of high grade paper daily, and costs $2000 per day to operate. Factory B produces 2 tons of low grade, 1 ton of medium grade, and 8 tons of high grade paper daily, and costs $4000 per day to operate. How many days should each factory operate to fill the orders at minimum cost? A. Write an objective function for cost and list the constraints that model the given situation.

B. Sketch a graph of the region determined by the constraints to find how many days each factory should operate in order to minimize cost.