Advanced Algebra & Trigonometry
Linear Programming Problems – Notes
Many business decisions are based on a complex set of inputs. For instance, a
company may have a set amount of materials (and/or time) to create their
products. The managers need to determine the ideal way to use these resources to
maximize their profit.
Example: A painter needs to mix 2 different shades of green paint, Spring Green and Flash
Green. He has 32 units of yellow tint and 54 units of green tint available to mix as much of these
2 shades as possible, without wasting the tint. Each gallon of Spring Green requires 4 units of
yellow tint and 1 unit of green tint. Each gallon of Flash Green requires 1 unit of yellow tint and
6 units of green tint.
Follow the Linear Program technique to find out how many gallons of Spring Green and Flash
Green the painter should mix to maximize the total amount of mixed paint.
1. Identify the variables.
x = # of gallons of Spring Green to be mixed
y = # of gallons of Flash Green to be mixed
2. Translate the constraints (limitations) in the problem to a system of
inequalities. Some constraints are maximums, others are minimums!
The limit of yellow tint is 32 units total:
The limit of green tint is 54 units total:
The # of gallons of both colors cannot be negative:
3. Graph the system of inequalities, shade the feasibility region. Label
all vertices of the feasibility region.
4. Write a function to be maximized or minimized (f(x,y)).
The points in the shaded region represent all the combinations of x and y
that are possible (or feasible), but which represents the optimum results?
We must, then write a function in terms of x and y to be maximized or
minimized. Re-read the problem and determine what the goal was. How
do the values of x and y relate to the goal? What function of x and y do
we want to maximize?
5. Substitute the ordered pair from each vertex into the optimization
function. Interpret the results.
The "Vertex Theorem" tells us that for the points in the feasibility region, only the
vertices will yield the maximum (or minimum if desired) values for the optimization
function. Plug the values from each vertex into the function and choose the best result.
Special situations on Linear Programming
If the constraints lead to no
region of overlap, the system is
infeasible.
y
10
If the feasibility region is an unbounded
Region, you can still use the bounded
vertices to determine the values to minimize
the function.
x
10
-1 0
-10