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Chapter 7
Linear Programming
You are taking an algebra test in which solving equations (questions of type A) are worth 10
points and story problems (questions of type B) are worth 15 points. It takes 3 minutes for each
question of type A and 6 minutes for each question of type B. The total time allowed for the test
is 60 minutes and you are not to answer more than 16 questions.
Assuming that all your answers are correct, how many items of each type should you answer in
order to get the best score?
Section 7.1 Systems of Linear Inequalities
Ex.
Given the linear inequality 2
(x, y)
5
10, determine if the ordered pairs are solutions.
Check
Solution?
(3, 1)
(1, -2)
(5, 0)
Finding/Representing ALL of the solutions
Case #1: Finding the solution set to a single linear inequality (in two variables)
Step #1: Graph the boundary line (solid if or , dashed if < or >)
Step #2: Choose a test point -typically, (0, 0), (0, 1), (1, 0) or (1,1) - and determine if it is a
solution to the inequality
Step #3: If the test point is a solution, shade the half-plane containing that point; otherwise,
shade the other half-plane.
Ex.
Graph
2
3
Graph2
5
10
Ex.
3
Graph
Graph
4
Case #2: Finding the solution set to a system of linear inequalities (in two variables)
Step #1: Graph each linear inequality separately (carefully keep track of the shading -- maybe
use colored pen/pencils or arrows)
Step #2: Determine the region(s) where the shading for all inequalities overlap. The solution
set may be bounded (solution set is confined to the boundary and interior of a
polygon) or unbounded.
Step #3: If necessary, find the corner points by solving the system of equations consisting of
the lines that form the points of intersection.
Ex.
Graph the solution set to the system of inequalities. Find any corner points, if they exist.
2
4
2
Ex.
Graph the solution set to the system of inequalities
2
Ex.
4
2
0
4
Graph the solution set to the system of inequalities. Find any corner points, if they exist.
2
2
0
0
1
Some additional notes…
1.
Sometimes a constraint may be redundant and does not affect the solution set
Ex. (from textbook)
2
2
0
2
6
2
2
(notice that line iii is not used to define the solution set) 2.
It is possible that a system of inequalities has no solution and we say the solution is the
"empty set" or ∅.
Ex. (from textbook)
5
3
3.
If all the points on a line segment between any two points in the solution set/ region of a
system of linear inequalities lie inside the set/region, we say the set/region is convex
Section 7.2
Finding An Optimal Value
Objective Function: A function for which we are trying to find the optimal value (the
minimum/maximum value).
Feasible Region: The solution set to a system of constraints (linear inequalities) and its corner
points (see Section 7.1)
Fundamental Theorem of Linear Programming
Given an objective function and feasible region
 If the objective function has an optimal value, then it will occur at one or more of the corner
points of the feasible region.
 If an optimal value occurs at two corner points (rare), it will also occur at any point on the
line segment that connects those corner points
 Whichever corner point yields the largest value for the objective function is the maximum
and whichever corner point yields the smallest value for the objective function is the
minimum.
 Sometimes a maximum or minimum value may not exist, depending on whether the feasible
region is bounded or unbounded.
Ex.
Find the maximum and minimum values of the objective function
where…
2
,
Corner
Points
8
6
0
3
2
3
2 ,
Ex.
Find the maximum and minimum values of the objective function
where…
3
3
,
5
2
Corner
Points
Ex.
0
,
3
4 ,
15
12
2
Find the maximum and minimum values of the objective function
where…
3
2
,
Corner
Points
2
9
7
8
0
3
4
Section 7.3
Ex.
Linear Programming Applications
You are taking an algebra test in which solving equations (question of type A) are worth
10 points and story problems (questions of type B) are worth 15 points. It takes 3 minutes
for each question of type A and 6 minutes for each question of type B. The total time
allowed for the test is 60 minutes and you are not to answer more than 16 questions.
Assuming that all your answers are correct, how many items of each type should you
answer in order to get the best score?
Objective Function
Constraints
Feasible Region and Corner Points
Corner
Points
Midwestern Mattresses has contracted to make at least 250 mattresses per week, which are to be
shipped to two stores, A and B. Store A has a maximum capacity of 140 mattresses, and Store
B has a maximum capacity of 165 mattresses. It costs $13 to ship a mattress to Store A and $11
to ship a mattress to Store B.
How many mattresses should be shipped to each store to minimize shipping costs?
Objective Function
Constraints
Feasible Region and Corner Points
(a, b)
C(a, b) =