Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 7.6 Linear Programming

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 7.6

Linear Programming


What You Will Learn

Linear Programming

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Linear ProgrammingIn a linear programming problem, there are restrictions called constraints.Each constraint is represented as a linear inequality.The list of constraints forms a system of linear inequalities.When the system of inequalities is graphed, often a feasible region is obtained.

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Linear ProgrammingThe points where two or more boundaries intersect are called the vertices of the feasible region.

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Objective FunctionThe objective function is the formula for the quantity K = Ax + By (or some other variable) that we want to maximize or minimize.The values we substitute for x and y determine the value of K.From the information given in the problem, we determine the real number constants A and B.

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Fundamental Principle of Linear Programming

If the objective function, K = Ax + By, is evaluated at each point in a feasible region, the maximum and minimum values of the equation occur at vertices of the region.

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Solving a Linear Programming Problem1. Determine all necessary

constraints.2. Determine the objective

function.3. Graph the constraints and

determine the feasible region.4. Determine the vertices of the

feasible region.7.6-7


Solving a Linear Programming Problem5.Determine the value of the

objective function at each vertex. The solution is determined by the

vertex that yields the maximum or minimum value of the objective function.

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Example 2: Modeling–Washers and Dryers, Maximizing Profit The Admiral Appliance Company makes washers and dryers. The company must manufacture at least one washer per day to ship to one of its customers. No more than 6 washers can be manufactured due to production restrictions. The number of dryers manufactured cannot exceed 7 per day.

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Example 2: Modeling–Washers and Dryers, Maximizing Profit Also, the number of washers manufactured cannot exceed the number of dryers manufactured per day. If the profit on each washer is $20 and the profit on each dryer is $30, how many of each appliance should the company make per day to maximize profits? What is the maximum profit?

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Example 2: Modeling–Washers and Dryers, Maximizing Profit SolutionLet x = the number of washers manufactured per day y = the number of dryers manufactured per day20x = the profit on washers30y = the profit on dryers P = the total profit

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Example 2: Modeling–Washers and Dryers, Maximizing Profit SolutionConstraintsNumber of appliances manufactured cannot be negative: x ≥ 0, y ≥ 0At least one washer per day: x ≥ 1No more than 6 washers per day: x ≤ 6No more than 7 dryers per day: y ≤ 7Washers cannot exceed dryers: x ≤ yx ≥ 0, y ≥ 0, x ≥ 1, x ≤ 6, y ≤ 7, x ≤ y

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Example 2: Modeling–Washers and Dryers, Maximizing Profit SolutionObjective function is the profit formula

P = 20x + 30y

Graph all the constraints to find the feasible region and the vertices.

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Example 2: Modeling–Washers and Dryers, Maximizing Profit Solution

The vertices are:(1, 1), (1, 7), (6, 7), and (6, 6)

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Example 2: Modeling–Washers and Dryers, Maximizing Profit SolutionCalculate the objective function at each vertex, P = 20x + 30y(1, 1): P = 20(1) + 30(1) = 50(1, 7): P = 20(1) + 30(7) = 230(6, 7): P = 20(6) + 30(7) = 330(6, 6): P = 20(6) + 30(6) = 300The company should manufacture 6 washers and 7 dryers. Profit is $330.

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 7.6 Linear Programming