Finite Math B: Chapter 3, Linear Programming – The Graphical Method 1

Chapter 3: Linear Programming The Graphical Method

Most realistic problems to not involve equations – they involve inequalities. For example: A business doesn’t want to produce EXACTLY 400 units of product (= 400), they need to produce AT LEAST 400 units of product, and hopefully more (≥400). A worker can run a machine for no more than 8 hours (≤8), etc. Linear Programming is a procedure used to OPTIMIZE a given factor that is governed by a system of inequalities that represent the situation. For example, a factory may want to MINIMIZE cost or a store may want to MAXIMIZE profits.

3.1 Graphing Linear Inequalities Algebra 2 Review: To graph a linear inequality in two variables 1. Graph the corresponding equation. If < or >, use a broken line. If ≤ or ≥ , use a solid line. 2. Decide which half-plane to shade. Method 1: If in the form y mx b or y mx b , shade the lower plane.

If in the form y mx b or y mx b , shade the upper plane.

Method 2: Choose a point not on the boundary line, and test back in the original inequality. TRUE result: shade the plane that contains the tested point. FALSE result: shade the plane that does NOT contain the tested point. Example 1: Graph 2 3 12x y

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Finite Math B: Chapter 3, Linear Programming – The Graphical Method 2 Example 2:

Graph 4 4x y

The solution to a SYSTEM OF INEQUALITIES, is the set of all points that satisfy all the inequalities at the same time. Graphically, this is the area of the graph where the shading for each inequality overlaps. Example 3: Graph the system

3 12

2

y x

x y

General Reminders about Lines: y mx b Line with slope m and y-intercept b

Positive slope: Negative slope: y b Zero slope (Horizontal Line)

x a Undefined/No slope (Vertical Line)

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Finite Math B: Chapter 3, Linear Programming – The Graphical Method 3 Example 4:

2

4 3

7

x

y

x y

A region consisting of the overlapping parts of two or more graphs of inequalities in a system is sometimes called the region of feasible solutions or just the feasible region. Sometimes a feasible region will be bounded, or enclosed on all sides by boundary lines. Sometimes a feasible region will be unbounded. A corner point is a point in the feasible region where two boundary lines intersect.

3.2 Linear Programming – Graphing Method - Linear Programming Terminology Objective Function: A function for which we are trying to find a MAXIMUM or MINIMUM value. Usually the objective function represents something like “profit” or “cost”. Constraints: A set of restrictions on the problem, represented by a system of linear inequalities. Feasible Region: Graphically, the set of all points that satisfy all of the constraints. (the overlap) Corner Points: The vertices of the feasible region, where two boundary lines intersect. Corner Point Theorem: If an optimum value (either a maximum or a minimum) of the objective function exists, then it will occur at one or more of the corner points of the feasible region.

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Finite Math B: Chapter 3, Linear Programming – The Graphical Method 4 Corner Point Theorem: Consider the feasible region graphed at the right. This represents all possible solutions to the system of inequalities. The corner point theorem tells us that IF any of these values (in the shading, on the border lines, etc) will produce and extreme value (a maximum or minimum) it will ALWAYS occurs at one of the corner points. Example 1: What are the maximum and minimum for the objective function: 2 5z x y subject to the constraints graphed at

the right?

Linear Programming Procedure 1. (If necessary) Write the objective function and all necessary constraints. 2. Graph the feasible region. 3. Identify all corner points. 4. Find the value of the objective function at each corner point. 5. For a bounded region, the solution is given by the corner point producing the optimum value of the objective function. 6. For an unbounded region, check that a solution actually exists. If it does, it will occur at a corner point.

Example 2: Maximize and Minimize

10z x y

Subject to:

4 12

2 0

2 6

6

x y

x y

y x

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Finite Math B: Chapter 3, Linear Programming – The Graphical Method 5 Example 3: Maximize

3 4z x y

Subject to:

2 4

2 4

0

0

x y

x y

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y

Special Notes: Lines that don’t intersect in a point that looks like (integer, integer): You will need to solve the system of equations to find the exact intersection point. You can use any of the following to find the solution to the system of equations: 1. Substitution/Elimination (from Algebra 1) 2. Gauss/Jordan (from Chapter 2) 3. Inverse Method (from Chapter 2) Horizontal/Vertical Lines You need to KNOW that horizontal lines have y = # equations and that vertical lines have x = # equations. Many linear programming problems are incorrectly solved due to incorrect graphing of horizontal lines. Examples: y > 0 x > 0 4 < y < 10 -3 < x < 5

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Finite Math B: Chapter 3, Linear Programming – The Graphical Method 6

3.3 Applications of Linear Programming Procedure 1. READ THE QUESTION CAREFULLY. 2. Underline the question and determine your variables. 3. Organize the data in a chart. 4. Follow “Linear Programming Procedure” from 3.2 Example 1: Set up each Linear Programming Problem. Write the constraints and the objective function. Do not solve. a) Mr. Trenga plans to start a new business called River Explorers, which will rent canoes and kayaks to people to travel 10 miles down the Clarion River in Cook Forest State Park. He has $45,000 to purchase new boats. He can buy the canoes for $600 each and the kayaks for $750 each. His facility can hold up to 65 boats. The canoes will rent for $25 per day and the kayaks will rent for $30 per day. How many canoes and how many kayaks should he buy to earn the most revenue? b) Certain animals in a rescue shelter must have a least 30 g of protein and at least 20 g of fat per feeding period. These nutrients come from food A, which costs 18 cents per unit and supplies 2 g of protein and 4 g of fat; and food B, which costs 12 cents per unit and provides 6 g of protein and 2 g of fat. Food B is bought under a long-term contract that requires that at least 2 units of B be used per serving. How much of each food must be bought to produce the minimum cost per serving?

3.2 Linear Programming Procedure 1. (If necessary) Write the objective function and all necessary constraints. 2. Graph the feasible region. 3. Identify all corner points. 4. Find the value of the objective function at each corner point. 5. For a bounded region, the solution is given by the corner point producing the optimum value of the objective function. 6. For an unbounded region, check that a solution actually exists. If it does, it will occur at a corner point.

Finite Math B: Chapter 3, Linear Programming – The Graphical Method 7 Example 2: Set up and Solve each Linear Programming Problem. a) A 4-H member raises goats and pigs. She wants to raise no more than 16 animals, including no more than 10 goats. She spends $25 to raise a goat and $75 to raise a pig, and she has $900 available for the project. The 4-H member wishes to maximize her profits. Each goat produces $12 in profit and each pig $40 in profit. How many of each type of animal should she raise?

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Finite Math B: Chapter 3, Linear Programming – The Graphical Method 8 b) A banquet hall offers two types of tables for rent: 6-person rectangular tables at a cost of $27 each and 10-person round tables at a cost of $46 each. You would like to rent the hall for a wedding banquet and need tables for 230 people. The room can have a maximum of 39 tables and the hall only has 15 rectangular tables available. How many of each type of table should be rented to minimize cost?

What is the minimum cost?

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