Applied Mathematics for Business and Economics

Lecture Note Linear Programming (LP)

Thus, the largest value of F is 25 at (3, 4), and the smallest value is 0 at the origin.

Example 3
What can be said about the largest and samllest values of the function

e following inequalities:

Fxy=+ 43

subject to th

7

57 4

,0

xy

xy

xy

5

+ ≥

+ ≥

≥

()0, 7•

()2, 5

(^) •
(^) (9, 0)
From the figure the feasible region R is unbounded (it cannot be contained in a circle)
and that its corner points are()0, 7 (2, 5) (9, 0). This suggests that there is a
smallest value of F and R but no largest value. Evaluating the corner points of R, we
obtain the following table:
Corner
Point
Value of
Fxy= 43 +
(0, 7)^21
(2, 5)^23
(9, 0) (^36)
Thus, the largest value of F doesn’t exist and the smallest value of F subject to the
given constraints is 21 which occurs at the point where x= 0 andy= 7.
Example 4
A farmer prepares feed for livestock by combining two grains. Each unit of the first
grain costs 20 cents and contains 2 units of protein and 5 units of iron, while each unit
of the second grain costs 30 cents and contains 4 units of protein and 1 unit of iron.
Each animal must receive at least 10 units of protein and 16 units of iron each day.
How many units of each grain should the farmer feed to each animal to satisfy these
nutritional requirements at the smallest possible cost?
Solution
For convenience, we construct the following table
Grain I Grain II Minimal Requirement
Protein 2 4 10
Iron 5 1 16
Cost 20 30
If we let
x: number of units of grain I fed daily to each animal
y: number of units of grain II fed daily to each animal