Iso-cost and iso-profit lines
Bounded regions
An example of the method on an unbounded region
§7.3–Linear Programming: Graphical Solutions
Tom Lewis
Fall Semester
2014
Iso-cost and iso-profit lines
Bounded regions
An example of the method on an unbounded region
Outline
Iso-cost and iso-profit lines
Bounded regions
An example of the method on an unbounded region
Iso-cost and iso-profit lines
Bounded regions
An example of the method on an unbounded region
Problem
Ace widget company makes two types of widgets: model A and
model B. It makes a profit of $4.00 per widget A and $3.00 per
widget B.
• Describe (by a graph) the production schedules which yield a
profit of $110.00.
• Describe (by a graph) the production schedules which yield a
profit of $240.00.
Iso-cost and iso-profit lines
Bounded regions
An example of the method on an unbounded region
Definition (Iso-profit and iso-cost lines)
The lines in our previous example are called iso-profit lines, lines of
constant profit. There is a related notion of iso-cost line, a line on
which the profit is constant.
Iso-cost and iso-profit lines
Bounded regions
An example of the method on an unbounded region
Problem
Maximize the function P = 3x + 4y subject to the constraints:
y + 2x 6 8, 3y + x 6 9, x > 0 and y > 0.
Iso-cost and iso-profit lines
Bounded regions
An example of the method on an unbounded region
Problem
Minimize the function C = 2x + 5y subject to the constraints:
x + 2y > 4, 3x + 5y 6 15, x > 0, y > 0.
Iso-cost and iso-profit lines
Bounded regions
An example of the method on an unbounded region
Theorem (Corner-Point Theorem)
If the feasible set is bounded, then the maximum and minimum
values of a function of the form Ax + By (profit or cost) will be
attained at the corner points of the region.
Iso-cost and iso-profit lines
Bounded regions
An example of the method on an unbounded region
Problem
Find the minimum value of C = 10x + 13y subject to the
constraints 2x + 3y > 6, x > 0 and y > 0.
Remark
In the previous example, the corresponding maximum value for C
does not exist.
Iso-cost and iso-profit lines
Bounded regions
An example of the method on an unbounded region
Problem
Completely solve the following linear programming problems from
§7.1:
• Problem #2
• Problem #8
• Problem #24