MATH 141 501
Section 3.2 Lecture Notes
Setting Up Linear Programming Problems: We have already seen examples where businesses use math to figure out how to make the most money
possible. (i.e. maximize their profit). This is one of the most common uses
of math in business, so it’s important that you get good at it! In this section,
we will be looking at setting up problems that involve finding the maximum or
minimum of a quantity subject to some constraints (inequalities).
Example 1: MotorCo makes a 100 horsepower motor and a 300 horsepower
motor. MotorCo’s profit is $100 for the 100 horsepower motor, and $120 for the
300 horsepower motor. Two types of machines are used in making the motors.
The 100 horsepower motor requires 20 minutes on Machine I and 10 minutes
on Machine II, while the 300 horsepower motor requires 10 minutes on Machine
I and 30 minutes on Machine II. There are 30 hours available on Machine I,
and 50 hours available on Machine II. How many engines of each type should
MotorCo make to maximize its profit?
Step 1: Make a table to represent the given information.
Machine I
Machine II
Profit (per Unit)
100 Horsepower
20 minutes
10 minutes
$100
300 Horsepower
10 minutes
30 minutes
$120
Time Available
1800 minutes
3000 minutes
Step 2: Declare variables.
Step 3: Determine the quantity to be maximized or minimized. This is
called the (objective function).
Step 4: Determine the constraints which the solution must obey.
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Linear Programming Problem
A linear programming problem consists of a linear objective function to
be maximized or minimized subject to certain constraints in the form of linear
equations or inequalities.
Example 2: Formulate but do not solve the following exercise as a linear
programming problem.
A town in Texas needs at least 1 million gallons of potable water per day.
The water can come from the local reservoir or from a pipeline to a nearby town.
The local reservoir can yield 500,000 gallons of potable water. The pipeline has
a maximum daily yield of 1 million gallons. An agreement between the two
towns says that the pipline must supply 600,000 gallons per day. The cost for
100,000 gallons of water from the reservoir is $200, while the cost for 100,000
gallons of pipeline water is $400. How much water should the town get from
each source to minimize the daily water costs for the city?
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Another Example
Example 3: Formulate but do not solve the following exercise as a linear
programming problem. (Note: The following question is very similar to one in
the homework.)
Aggie Finance has a total of $2.4 million earmarked for homeowner loans
and business loans, where x represents homeowner loans in millions of dollars
and y represents business loans in millions of dollars. Homeowner loans give
a 8% annual rate of return. Business loans yield a 12% annual rate of return.
Management is worried about the perception that favors businesses over individuals, so it wants the total amount of homeowner loans to be at least four times
as much as the total amount of business loans. Determine the total amount
of loans of each type Aggie Finance should offer to maximize its returns P (in
millions of dollars).
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