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Linear Programming

Linear Programming. An Application of Inequalities in Two Variables (day 2). Linear Programming.

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Linear Programming

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  1. Linear Programming An Application of Inequalities in Two Variables (day 2)

  2. Linear Programming Decision makers in many fields, including business and science, try to find ways to allocate resources in order to maximize profits or productivity and to minimize cost. The quantity to be maximized is represented by a function (equation) called the objective function. The available resources, and the restrictions placed on them, can be represented by a system of inequalities called constraints.

  3. Linear programming problems involve maximizing or minimizing an objective function by subjecting it to linear constraints. The graph of a linear system of constraints is called the feasible region. Any point in this region satisfies each of the constraints. Mathematicians have proved that if there is a maximum or minimum value of the objective function, it will occur at a vertex of the feasible region. (Prentice Hall Algebra 2, p. 182)

  4. Example 1: Find the maximum and minimum value. • P = x + 4y • (6, 6) • (6, 10) • (8, 8) • (8, 6)

  5. Example 2: If the profit is represented by P = x + 3y, find the maximum profit under these constraints.

  6. Example 3: Jim Olsen makes and sells gourmet food items. He makes two types of salad dressing, garlic and tofu. Each gallon of garlic dressing requires 2 quarts of oil and 2 quarts of vinegar. Each gallon of tofu dressing requires 3 quarts of oil and 1 quart of vinegar. Jim makes a $3 profit on each gallon of garlic dressing and a $2 profit on each gallon of tofu dressing. He has 18 qt of oil and 10 qt of vinegar on hand. How many gallons of each type of dressing should he make to maximize his profits?

  7. Organize the Information • Let x = # gallons of garlic dressing • Let y = # gallons of tofu dressing

  8. Write the Objective Function Total profit = profit on garlic + profit on tofu P = 3x + 2y

  9. Write the Constraints Quarts of oil is no more than 18. Quarts of vinegar is no more than 10. Gallons of garlic dressing is greater than or equal to 0. Gallons of tofu dressing is greater than or equal to 0.

  10. Graph and Shade the Feasible Region

  11. Interpret the Results • The maximum value occurs at a vertex of the feasible region. Evaluate the objective function at each vertex. • To maximize profits, Jim should make gal of garlic dressing and gal of tofu.

  12. Homework: • Page 280 #30, 32, 35, 40, 42

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