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Linear Programming Pre-Calc Section 3.4 Running a profitable business requires a careful balancing of resources (for example, peoples’ time, materials, and machine availability). A manager must choose the best use of these resources. Often the range of possible choices can be described by a set of linear inequalities, called—constraints. In most situations the number of alternative solutions to the constraints is so great that it is hard to find the best one. Linear Programming -- a process which makes many previously impossible problems solvable.

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Example 1: suppose that a small TV manufacturing company produces Flat Screen and ‘big Screen’ TV’s, using three different machines, A, B, and C. The table below shows how many hours are required on each machine per day in order to produce a ‘Flat Screen’ TV or a Big Screen’ TV. Hours Machine | Flat Screen | Big Screen | Available | A | 1 h | 2 h | 16 | A | 1 h | 2 h | 16 | B | 1 h | 1 h | 9 | B | 1 h | 1 h | 9 | C | 4 h | 1 h | 24 | C | 4 h | 1 h | 24 | 1 st : let x = Number of Flat Screen TV’s y = Number of Big Screen TV’s Now using the information in this chart along with some common sense, we can come up with the following:

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a. x > 0 (The number of TV’s cannot be negative) b.y > 0 (Now machine A needs 1 hour for each Flat Screen TV and 2 hours for each Big Screen TV. Thus for ‘x’ Flat Screen’s c. x+2y < 16 and ‘y’ Big Screen’s Machine A needs ‘1x + 2y’ hours. Since this machine is available for at most 16 hours a day, x + 2y < 16) (The last two inequalities are similar to the one above. Machines B and C are d.x+y < 9 available for at most 9 hours and 24 e.4x+ y < 24 respectively) Now sketch the graph of all ‘5’ inequalities ‘simultaneously Check out graphs ‘a-e’ located at bottom of page 109)

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Now suppose that the manufacturing company described earlier makes a $60 profit on ‘Flat Screen’s’ and a $40 profit on each ‘Big Screen’. How many ‘Flat Screens’ and how many ‘Big Screens’ should be produced each day to maximize the profit? The profit for x Flat Screens and y Big Screens P = 60x + 40y (dollars) To save us the time of plugging in all possible (x,y) Ordered pairs in our ‘feasible region’ research has been Done to show that a maximum or a minimum value will Always occur at a ‘corner point’ Therefore all we need to do is check the ‘5’ corner (x,y) Points to realize the maximum! Notice figure on pg 110 and the ‘blue’ profit lines shown. Also notice what each profit is equal to, so you can identify when the maximum profit is reached at (5,4)

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Example 2 Minimizing a cost Every day Rhonda Miller needs a dietary supplement of 4 mg of vitamin A, 11 mg of vitamin B, and 100 mg of Vitamin C. Either of two brands of vitamin pills can be used: Brand X at 6 cents a pill or Brand Y at 8 cents per pill. The chart below shows that a Brand X pill supplies 2 mg of Vitamin A, 3 mg of vitamin B, and 25 mg of vitamin C. Likewise, a Brand Y pill supplies 1, 4, and 50 mg respectively. How many pills of each brand should she take each day in order to satisfy the minimum daily need most economically? | | Brand X | Brand Y | Minimum daily need| | Vitamin A | 2 mg | 1 mg | 4 mg | | Vitamin B | 3 mg | 4 mg | 11 mg | | Vitamin C | 25 mg | 50 mg | 100 mg | |Cost per pill | 6 cents | 8 cents | |

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Solution: Let x = number of Brand X pills y = number of Brand Y pills System of inequalities: x > 0 y > 0 2x + y > 4 3x + 4y > 11 x + 2y > 4 Determine the feasible region by graphing the system of inequalities. The cost (in cents) can be expressed by the equation C = 6x + 8y. Evaluate the cost at each corner point.

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Applications of Linear Programming

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