Question 1: What is a linear programming problem?
Linear programming is a strategy for maximizing or minimizing a linear function of
several variables. This function is called the objective function and takes the form
z  a1 x1  a2 x2    an xn . Although this equation may look imposing, we can break it down into a simple pattern.
Each term on the right hand side consists of a constant called an multiplied by a
decision variable xn . The names of the constants and variables are irrelevant, but the
constant an always multiplies the decision variable xn raised to the first power. The
decision variables are the variables that represent the quantities that may be varied.
The variable z represents the quantity being maximized or minimized.
A linear programming problem consists of an objective function and a system of
inequalities that defines acceptable values for the variables x1 through xn .
Linear Programming Problem
A linear programming problem has the form
Maximize or Minimize z  a1 x1  a2 x2    an xn
subject to constraints of the form
b1 x1  b2 x2    bn xn  c or b1 x1  b2 x2    bn xn  c
and x1  0, x2  0,  , xn  0 .
In a linear programming problem, the constants a1 , , an , b1 ,, bn , and c are real
numbers.
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The values of the decision variables that optimize (maximize or minimize) the value of
the objective function are called the optimal solution of the linear programming
problem.
Businesses typically try to either maximize profit or minimize cost by varying the values
of the decision variables. With the addition of an objective function, the craft brewery
problem can be written as a linear programming problem. Suppose that each barrel of
pale ale yields $100 of profit and each barrel of porter yields $80 of profit. We can
multiply the profit per barrel times the number of barrels for each type of beer to get the
profit for each type of beer. The profit from x1 barrels of pale ale is
Profit from pale ale 
100 dollars
 x1 barrels  100 x1 dollars
1 barrel
and the profit from x2 barrels of porter is
Profit from porter 
80 dollars
 x2 barrels  80 x2 dollars
1 barrel
The total profit P is the sum of the profit from each type of beer,
P  100 x1  80 x2
Although this equation does not utilize z as a variable, it is the objective function for a
craft brewery. The names of the variables are not important. We could just as easily
used z instead of P or Q1 and Q2 instead of x1 and x2 . The key ingredient of a linear
objective function is that the decision variables are raised to the first power, multiplied
by a constant and then added together.
With this objective function in hand, we can write out the linear programming problem
for the craft brewery:
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Maximize P  100 x1  80 x2
subject to
x1  x2  50, 000
69.75 x1  85.25 x2  4, 000, 000
23.8 x1  10.85 x2  1, 000, 000
x1  0, x2  0
In the next question, we’ll examine how this problem can be solved using a graph of
the solution of the system of inequalities.
Example 1
Find the Linear Programming Problem
Some breweries contract with other breweries to brew a portion of their
annual output. By doing this, they avoid the high fixed costs associated
with setting up and operating a brewing facility. One brewery, Patrick
Henry Brewing, contracts with two other breweries to produce an
American ale. It costs the brewery $100 per barrel to produce the beer
at the first contract brewery and $125 per barrel to produce the beer at
the second contract brewery. To satisfy orders from distributors,
Patrick Henry Brewing must contract for a total of at least 10,000
barrels per month. In addition, the amount that is contracted from the
second brewery must be at least 25% of the amount contracted from
the first brewery. However, labor agreements require that the amount
contracted from the second brewery cannot exceed the amount
contracted from the first brewery.
Write out the linear programming problem for determining how many
barrels should be produced from each contract brewer to ensure that
the total cost is minimized?
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Solution To begin, identify what we are trying to find and define
variables to reflect the unknowns. Since Patrick Henry Brewing needs
to know how many barrels of American ale to produce at each of the
contract brewers, define the variables as follows:
Q1: Number of barrels produced at contract brewery 1
Q2 : Number of barrels produced at contract brewery 2
Producing these quantities costs Patrick Henry Brewing $100 per barrel
at brewery 1 and $125 per barrel at brewery 2. The cost to produce Q1
barrels at contract brewery 1 is
Cost at contract brewery 1 
100 dollars
 Q1 barrels  100Q1 dollars
1 barrel
and the cost to produce Q2 barrels at contract brewery 2 is
Cost at contract brewery 2 
125 dollars
 Q2 barrels  125Q2 dollars
1 barrel
The total cost C is the sum of the individual costs from each contract
brewery,
C  100Q1  125Q2
We want to minimize the total cost subject to any constraints on
production.
The inequalities are signaled by phrases like “at least” and “cannot
exceed”. First we note that the brewery must contract for at least
10,000 barrels per month. This means that in any month,
Total
amount
contracted


  10, 000
Q1  Q2
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The phrase “at least” means that the total amount contracted must be
greater than or equal to 10,000. Substituting the sum of the variables
for the total amount contracted yields
Q1  Q2  10, 000 .
Similarly, the amount contracted from brewery 2 must be at least 25%
of the amount contracted from brewery 1 or

Q2

Amount from
brewery 2
0.25Q1

25% of the amount
from brewery 1
Since the amount contracted from the second brewery cannot exceed
the amount contracted from the first brewery, we can also write
Q2  Q1
Putting the objective function together with the inequalities (including
non-negativity constraints) gives the linear programming problem
Minimize C  100Q1  125Q2
subject to
Q1  Q2  10, 000
Q2  0.25Q1
Q2  Q1
Q1  0, Q2  0
The non-negativity constraints are needed since the number of barrels
of beer from each contract brewery cannot be negative. We’ll find the
solution to this linear programming problem in the next question.
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