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2.7 Linear Programming

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Linear programming - Certain constraints exist or are placed upon the variables and some function of these variables must be maximized or minimized. The constraints are written as a system of linear inequalities.

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**LP procedure: Define variables.**

Write the constraints as a system of inequalities. Graph the system and find the coordinates of the vertices. Write an expression whose value is to be maximized or minimized. Substitute values from the coordinates of the vertices into the expression. Select the greatest or least result.

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Ex 1 The profit on each set of cd’s that is manufactured is $8. The profit on a single CD is $2. Machines A and B are used to produce both types of CD’s. Each set takes nine minutes on Machine A and three minutes on Machine B. Each single takes one minute on Machine A and one minute on Machine B. If Machine A is run for 54 minutes and Machine B is run for 42 minutes, determine the combination of cd’s that can be manufactured during the time period that most effectively generates profit within the given constraints.

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**Different types of regions**

Infeasible – when the constraints cannot be satisfied simultaneously Unbounded – an optimal solution may not exist. (may not have a max or min.) Alternate optimal solutions – two or more optimal solutions. When the graph of the function to be maximized or minimized is parallel to one side of the polygonal convex set.

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Ex 2 A manufacturer makes widgets and gadgets. At least 500 widgets and 700 gadgets are needed to meet minimum daily demands. The machinery can produce no more than 1200 widgets and 1400 gadgets per day. The combined number of widgets and gadgets that the packaging department can handle is 2300 per day. If the company sells both widgets and gadgets for $1.59 each, how many of each item should be produced in order to maximize profits?

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