# Linear Programming and Minimization of Cost-Graphical Method:
**Linear programming graphical method** can be applied to **minimization problems** in the same
manner as illustrated on
maximization example page.
An example can help us explain the procedure of
minimizing cost using linear programming graphical method. ##
Example:
Assume that a pharmaceutical firm is to produce
exactly 40 gallons of mixture in which the basic ingredients, x and y, cost
$8 per gallon and $15 per gallon, respectively, No more than 12 gallons of x
can be used, and at least 10 gallons of y must be used. The firm wants to
minimize cost.
The cost function objective can be written as:
C) = 8x + 15y
C = Cost
The problem illustrates the three types of
constraints, =, ≤, and ≥, as follows:
x + y = 40
x ≤ 12
y ≥ 10
The optimum solution is obvious. Since x is
cheaper, as much of it as possible should be used, i.e., 12 gallons. Then
enough y, or 28 gallons, should be used to obtain the desired total quantity
of 40 gallons. ##
Graphical Method:
The constraints define the solution space
when they are plotted on the graph below:
**Graph**
The solution space indicate the area of feasible
solution represented by the line AB. Any combination of x and y falls within
the solution space (line AB) is a feasible solution. However, the best
feasible solution is found at one of the corner points, A or B.
Consequently, the corner points must be examined to find the combination
that minimizes cost, i.e., $8x + $15y.
A--------(x = 0, y = 40); $8(0) + $15(40) =
$600cost
B----(x = 12, y = 28); $8(12) + $15(28) =
$516cost
To minimize cost, the company should use 12
gallons of x and 28 gallons of y at a total cost of $516. |