Linear Programming: Introduction

Topic: Optimizing linear systems (Section 5.6 of the textbook, p 540)

Linear programming is the process of taking various linear inequalities relating to some situation, and finding the "best" value obtainable under those conditions. A typical example would be taking the limitations of materials and labor, and then determining the "best" production levels for maximal profits under those conditions.

In "real life", linear programming is part of a very important area of mathematics called "optimization techniques". These fields of study (or at least the applied results of it) are used every day in the organization and allocation of resources. These "real life" systems can have dozens or hundreds of variables, or more. In algebra, though, you'll only work with the simple (and graphable) two-variable linear case.

The general process for solving linear-programming exercises is to graph the inequalities (called the "constraints") to form a walled-off area on the x,y-plane (called the "feasibility region"). Then you figure out the coordinates of the corners of this feasibility region (that is, you find the intersection points of the various pairs of lines), and test these corner points in the formula (called the "optimization equation") for which you're trying to find the highest or lowest value. If the function has a maximum or a minimum value, then it must occur at one of these vertices of the feasible region.

EXAMPLE: Find the maximal and minimal value of z = 3x + 4y subject to the following constraints:

The three inequalities in the curly braces are the constraints. The area of the plane that they mark off will be the feasibility region. The formula "z = 3x + 4y" is the optimization equation. I need to find the (x, y) corner points of the feasibility region that return the largest and smallest values of z.

My first step is to solve each inequality for the more-easily graphed equivalent forms: It's easy to graph the system:

To find the corner points -- which aren't always clear from the graph -- I'll pair the lines (thus forming a system of linear equations) and solve:

1 1 y = –( /2 )x + 7 y = –( /2 )x + 7 y = 3x y = 3x y = x – 2 y = x – 2

1 1 –( /2 )x + 7 = 3x –( /2 )x + 7 = x – 2 3x = x – 2 –x + 14 = 6x –x + 14 = 2x – 4 2x = –2 14 = 7x 18 = 3x x = –1 2 = x 6 = x y = 3(–1) = –3 y = 3(2) = 6 y = (6) – 2 = 4 corner point at (2, 6) corner point at (6, 4) corner point at (–1, –3)

So the corner points are (2, 6), (6, 4), and (–1, –3). Since one of these three points will have to be the maximum or minimum, to solve you would plug these values into the equation: z = 3x + 4y.

(2, 6): z = 3(2) + 4(6) = 6 + 24 = 30 (6, 4): z = 3(6) + 4(4) = 18 + 16 = 34 (–1, –3): z = 3(–1) + 4(–3) = –3 – 12 = –15

Then the maximum of z = 34 occurs at (6, 4), and the minimum of z = –15 occurs at (– 1, –3). Steps: 1. Solve each for the y-intercept form of a line for easier graphing. 2. Find the vertices. 3. Replace each ordered pair for (x,y) in the objective function to find the minimum(s) and the maximum.

Directions: Find the minimum and maximum of the objective functions subject to the given constraints.

1. Objective function C=3 x + 2 y

3x+ 4 y 20 3x- y 5 Constraints: x 0 y 0

2. Objective functionC=6 x - 2 y

x+ y 9 4x+ y 12 Constraints: x 0 y 0

3. Objective function: C= x + 3 y

x+ y 5 Constraints: x 1 y 2

4. Objective function: C=2 x + 5 y

3x- y 5 24 x- y 6 Constraints: x 0 y 0 5. Objective function: C= -2 x + y y- x 0 Constraints: x 2 y 1

6. Objective function: C=3 x - y

-9x - 4 y � 16 5x- 3 y 35 Constraints: x 0 y � 5

x 0 y 0 Constraints: 3x- 2 y 6 y�+ x 7

x+ y 6 Constraints: x 8 y 5

9. Objective function: T=3 x + 2 y

x吵0, y 0 Constraints: 2x+ y 8 x+ y 4 10. Objective function: T=4 x + y

x吵0, y 0 Constraints: 2x+ 3 y 12 x+ y 3

x吵0, y 0 2x+ 3 y 12 Constraints: 3x+ 2 y 12 x+ y 2

x吵0, y 0 x+ 3 y 6 Constraints: x+ y 3 x+ y 9

x吵0, y 0 x+ y 7 Constraints: 2x+ y 10 2x+ 3 y 18

x吵0, y 0 2x+ y 10 Constraints: x+ 2 y 10 x+ y 10

The dreaded story problems…(but only three)....

15. Food and clothing are shipped to survivors of a natural disaster. Each carton of food will feed 12 people, while each carton of clothing will help 5 people. Each 20- cubic-foot box of food weighs 50 pounds and each 10-cubic-foot box of clothing weighs 20 pounds. The commercial carriers transporting food and clothing are bound by the following constraints:  The total weight per carrier cannot exceed 19,000 pounds.  The total volume must be less than 8000 cubic feet.

How many cartons of food and clothing should be sent with each plane to maximize the number of people who can be helped?

16. You are about to take a test that contains computation problems worth 6 points each and word problems worth 10 points each. You can do a computation problem in two minutes and a word problem in four minutes. You have 40 minutes to take the test and may answer no more than 12 problems. Assuming you answer all the problems attempted correctly, how many of each type of problem must you answer to maximize your score? What is the maximum score?

17. A large institution is preparing lunch menus containing foods A and B. The specifications for the tow foods are given in the following table:

Units of Units of Units of Food Fat per Carbohydrates Protein Ounce Per Ounce Per Ounce

A 1 2 1 B 1 1 1 Each lunch must provide at least 6 units of fat per serving, no more than 7 units of protein, and at least 10 units of carbohydrates. The institution can purchase food A for $0,12 per ounce and food B for $0.08 per ounce. How many ounces of each food should a serving contain to meet the dietary requirements at the least cost?