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Math 1313
Section 2.1
SECTION 2.1 – Solving Linear Programming Problems
Definitions:
An objective function is maximum P(x) = 3x + 2y or min C(x) = 4x + 8y
Constraints are a :
5x + 3y ≤ 120
2x + 6y ≤ 60
Consider the following figure which is associated with a system of linear
inequalities:
P( x ) = 3 x + 2 y
x+y≤4
St:
2x + 5y ≤ 80
x, y ≥ 0
Definitions:
The set S is called a feasible set. Each point in S is a candidate for the solution of
the problem and is called a feasible solution.
The point(s) in S that optimizes (maximizes or minimizes) the objective function
is called the optimal solution.
Maximum
Minimum
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Math 1313
Section 2.1
Problem with mixed constraints so has this type of feasible region. In mixed
constraint problem has both maximum and minimum possible solutions.
The Method of Corners
1. Graph the feasible set.
2. Find the coordinates of all corner points (vertices) of the feasible set.
3. Evaluate the objective function at each corner points.
4. Find the vertex that renders the objective function a maximum (minimum).
If there is only one such vertex, then this vertex constitutes a unique solution to
the problem. If the objective function is maximized (minimized) at two adjacent
corner points of S, there are infinitely many optimal solutions given by the
points on the line segment determined by these two vertices.
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Math 1313
Section 2.1
Example 1: How to find corner points and solve:
=
+
Subject to:
+
≤
+ ≤
≥ , ≥
Use the corner points the find the maximum value for this linear programing
problem:
Example 2: How to find corner points and solve:
=
Subject to:
+
+
≥
+
≥
≥ , ≥
Use the corner points the find the minimum value for this linear programing
problem:
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Math 1313
Section 2.1
Example 3: How to find corner points and solve:
=
+
Subject to:
+ ≤
2 +
≥
≥ , ≥
Find the maximum value:
Find the minimum value:
Example 4: Solve the problem by giving the optimal solution.
Q = 1200x + 100y
40x + 8y ≥ 400
Subject to:
3x + y ≤ 36
x, y ≥ 0
a. What is the maximum value?
b. What is the minimum value?
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Math 1313
Section 2.1
Example 5: Maximize the following Linear Programming Problem.
Minimize C = 18x + 12y
Subject to: 2x +6 y ≥ 30
4 x + 2 y ≥ 20
x≥0
y≥2
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