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.
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.
The constraints define the solution space when they are plotted on the graph below:
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.