High Step Shoes
A Linear Programming Problem
Airheads
Grounded
Linear programming is a method often used to solve large, complicated problems. These
problems often require a manager to determine how to use the company’s limited resources most
efficiently.
In order to see how linear programming is used to solve such problems, we are going to
investigate a problem experienced at the High Step Shoe Factory. The High Step Shoe problem
is just a fraction of the size of those encountered in the real world.
Step 1: Understanding the problem
Mr. B. Ball, director of manufacturing for the High Step Sports Shoe Corporation, wants to
maximize the company’s profits. The company makes two brands of sport shoe, Airheads and
Groundeds. The company earns $10 profit on each pair of Airheads and $8.50 profit on each
pair of Groundeds. The steps in manufacturing the shoes include cutting the materials on a
machine and having workers assemble the pieces into shoes.
Decision Variables
A=
Represent quantities that a manager can change.
Example: The number of shoes to be made.
G=
Mr. Ball’s goal is to make the most money or maximize his profits. Using A and G, write a
function to model the profit.
Objective Function
Profit = _____ A + _____ G
The equation represents the goal of either
maximizing profit or minimizing cost.
This function is called the objective
function.
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Step 2. Constraints: A System of Inequalities
The number of machines, workers, and factory operating hours put constraints on the number of
pairs of shoes that the company can make.
Constraints
Limitations created by scarce resources (time, equipment, etc.).
They are expressed algebraically by inequalities.
High Step Sports Shoe Corporation has the following constraints: 6 machines that cut the
materials, 850 workers that assemble the shoes, and a 40-hour workweek for the assembly plant.
A. Machine Cutting Constraint Inequality
Each hour, each cutting machine can do 50 minutes of work. How many minutes of work
can 6 machines do in a 40 hour work week?
_____ machines X _____ min. X _____ hr. = __________ min./wk.
hr.
wk.
Each pair of Airheads requires 3 minutes of cutting time while Groundeds require 2 minutes.
The inequality representing the constraint of the cutting machines is:
_________________________
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Why do we use ≤ 12,000 min. instead of = 12,000 min.?
B. Worker Assembly Constraint Inequality
How many minutes of work can 850 assembly workers do in a week?
_____ workers x _____hr/wk = ________ hr /wk
Each worker takes 7 hours to assemble a pair of Airheads and 8 hours to assemble a pair of
Groundeds. The inequality modeling the amount of time it takes to assemble the shoes for
the week is:
2
_____________________________
C. Other Constraints
The number of pairs of shoes that High Step manufactures is never negative,
but could it be zero? Why?
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Therefore, we have two more constraints on our variables, A and G.
3
A
G
4
Step 3: Graphing a System of Inequalities – Feasible Region
Feasible Region
Our system of constraint inequalities is:
1
Area containing all the points that
satisfy the constraints.
2
3
4
Values of A and G that satisfy all of the constraint inequalities are called feasible. The set of all
such feasible points is called the feasible region.
Graph the constraints on the grid below. Graph each line associated with an inequality by
finding that line’s x and y intercepts.
x-intercept
y-intercept
1
2
Label each axis.
Label each line with its equation.
Label the intercepts.
All possible solutions
to the problem lie in
the feasible region or
on the boundary.
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Step 4: Best Production Plan: Searching for the Optimal Solution
The best solution for this problem gives the company a maximum profit. The process of
determining this best solution is called optimizing, and the solution itself is called the optimal
solution.
Optimum
The optimal solution yields the best solution,
i.e., the most profit or the least cost.
1. One way to determine the optimal solution is to try all the answers.
a. Pick three points inside the feasible region and note them in the table below.
b. Test your points in the profit equation.
A = Airheads
1000
G = Grounded
3000
Profit =
10(1000) + 8.5(3000) = 35,5000
Compare your answers with other students to see who has the most profit. Is it possible to
figure the profit for each point in the feasible region? Why or why not?
The Corner Principle
2. Fortunately, there is an easier way.
Find all the corner points on the graph (points of
intersection) and test them in the profit equation.
The optimal solution will
always lie on a corner of
the feasible region.
Find the points of intersection by solving the system of inequalities. You may do this by
hand or using the graphing calculator. Label the points of intersection. These are called
corner points.
A = Airheads
0
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G = Grounded
Profit =
0
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Which point gives you the largest profit? ____________
What is the maximum profit? ____________
How many of each type of shoe should be produced?
Why does the Corner Principle work?
Turn the profit equation around to give 10A + 8.5 G = P. Substitute different quantities
for the profit, such as, $25,000, $30,000, $43,300, $45,000 and $50,000. Put in slopeintercept form and graph on your calculator.
Slope-intercept form
10A + 8.5G = 25,000
10A + 8.5G = 30,000
10A + 8.5G = 43,300
10A + 8.5G = 45,000
10A + 8.5G = 50,000
Solve each equation for G. What is the slope of each line? ____________
What is true about all these lines? ______________________
Do any of these lines fall in the feasible region? ________
Where will the line representing the optimal solution intersect the feasible
region? ________________________________________________
This is not a formal proof (we need some higher mathematics), but it shows that the profit
line that intersects the feasible region in just one point represents the maximum profit.
Adapted from: “High Step Shoes Blues: An Application of Systems of Linear Inequalities for High School Mathematics Classes
from Algebra I to Algebra II,” by Kenneth Chelst, Ph.D. and Tom Edwards, Ph.D., Wayne State University, Detroit, MI,
November, 1997, with minor changes.
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