5.5
Linear Programming
Objectives:
After completing this section, you should be able to:
1.
Solve linear programming problems;
2.
Use linear programming to model and solve reallife problems.
In Section 4, many of the problems we solved are related to the general type of problems called
linear programming problems.
Linear programming is a mathematical process that has been developed
to help management in decision making, and it has become one of the widely used tools in a process
called
optimization
.
In this section we will study linear programming as a strategy for finding the
optimal value
–
either maximum or minimum of a quantity subject to certain
constraints.
We will use an
intuitive graphical approach based on the techniques we discussed in Section 4 to illustrate this process
for problems involving two variables.
George Dantzig (1914
), an American mathematician, was the first to formulate a linear
programming problem in 1947 and introduced a solution technique, called
simplex method,
that finds the
optimal solution of complex linear programming problems involving thousands of variables and
inequalities. This method does not rely on graphing and is wellsuited for computer implementation.
Linear Programming Model
A Linear Programming problem
is concerned with finding the
optimal value
(maximum or
minimum value) of a linear
objective function
of the form
z = ax + by
where the
decision variables
x
and
y
are subject to
problem constraints
stated as a system of
linear inequalities
and to
nonnegative constraints
x, y ≥
0.
The set of points satisfying the system of linear inequalities
and the nonnegative constraints is
called the set of
feasible solutions
for the problem.
The graph of this set is the
feasible region
.
Any point in the feasible region that produces the optimal value of the objective function over
the feasible region is called an
optimal solution.
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Unit 5. Systems of Equations and Inequalities
Section 5
page 2
_____________________________________________________________________________________
Example 5.5.1
Linear Programming problem
Find the maximum value of
z =
2
x
+ 3
y
subject to
0
0
21
2
3
25
5
2
y
x
y
x
y
x
Because every point in the feasible region satisfies each constraint, the question is which of those
points will yield a maximum value of
z?
We cannot check all of the feasible solutions to see which one
results in the largest value of
z.
Fortunately, a simple way has been developed to find this optimal
solution.
Using advanced techniques, it can be shown that
If the feasible region is bounded, then one or more of the corner points of the feasible
region is an optimal solution to the problem.
This means that
you can find the maximum value of
z
by testing
z
at each of the corner points
and choose
the corner point that produces the largest value of
z.
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 Spring '11
 dikopaalam
 Math, Linear Programming, Optimization, objective function, niacin, Aling Maring, Theorem of Linear Programming, vitamin C.

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