Sec3 - Lecture notes CHAPTER 3 PDF

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3.3...

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Section 3.3 - Method of Corners Any point in the solution set (also called feasible region) is called a A feasible solution that optimizes the objective function is called an Theorem 1. If the feasible region of a linear programming problem is optimal solution must occur at one of the .

. . , then the

Example 1. Find the maximum and minimum value of the objective function P = 2x + y on the feasible region S .

Finding optimal solution using method of corners: 1. 2. 3. 4.

Graph the constraints and determine the solution set. If the solution set is bounded, find the corner points. Evaluate the objective function at each corner point. Determine the maximum (or minimum) value.

Example 2. Maximize P = 4x + 3y subject to the constraints: 2x + y ≤ 10,

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2x + 3y ≤ 18,

x ≥ 0,

y ≥ 0.

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Example 3. Minimize C = 10x + 15y subject to the constraints x + y ≤ 10,

Theorem 2. If the solution set is

3x + y ≥ 12,

x ≥ 0,

y ≥ 3.

, then an optimal solution may or may not exist.

If the objective function is P = ax + by, where a, b are the constraints include x ≥ 0, y ≥ 0, then P has a And P has an maximum value.

numbers and value at one of the corners.

Example 4. Find the maximum and minimum value of P = x + 2y, subject to the constraints: −2x + y ≤ 4,

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x + y ≥ 2,

x ≥ 0,

y ≥ 0.

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Example 5. You manage an ice cream factory that makes two flavors: Vanilla and Mocha. Each cup of Vanilla requires 2 eggs and 3 cups of cream. Each cup of Mocha requires 1 egg and 4 cups of cream. You have in stock 500 eggs and 1200 cups of cream. You make a profit of $3 on each cup of Vanilla and $5 on each cup of Mocha. Your goal is to make as much profit as possible with the available material. 1. Set up the linear programming problem to determine the number of cups of each flavor that you should make.

2. Graph the feasible region.What is the maximum profit you can make?

3. How many cups of each flavor should you make in order to earn the largest profit?

4. Are there any leftover resources?

Remark: In some problems, there might be NO feasible region! (Try: x ≥ 0, y ≥ 5, x + y < 2). In this case, we say that the linear programming problem is . Highly Suggested Homework Problems: 3, 5, 15, 23, 25, 29, 31, 35, 39, 45.

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