Page 3 - LESSON NOTES

The values of Z at these corner points are as follows:

It can be seen that the value of Z at points A and B is same. If we take any other point such as
(2, 2) on

line x + 2y = 6, then Z = 6

Thus, the minimum value of Z occurs for more than 2 points.

Therefore, the value of Z is minimum at every point on the line x + 2y = 6

Linear Programming Problem and its Mathematical Formulation:
Formulation of an LPP refers to translating the real-world problem into the form of
mathematical equations which could be solved. It usually requires a thorough understanding of
the problem.

Let us take an example to illustrate the formulation of linear programming problem in various
different situations.

Example: 3

Two food stuffs F1 and F2 contain vitamins A, B, C. The minimum daily requirements of
these vitamins for a certain diet are 3 mg of A, 50 mg of B and 40 mg of C. One unit of the food -
stuff F1 contain 1 mg of A, 25 mg of B and 10 mg of C whereas one unit of the food-stuff
F2 contains 1 mg of A,10 mg of B and 20 mg of C. The cost of one unit food-stuff F1 is Rs 1 and
that of F2 is Rs 2. Formulate the problem as a linear programming problem.

Solution:
Let x units of F1 and y units of F2 are used in the diet.
Since 1 mg of F1 contains 1 mg of A and one unit of F2 contains 1 mg of A and the minimum
requirement of A is 3 mg.
Therefore, x + y ≥ 3

Similarly for vitamins B and C, we have

25x + 10y ≥ 50 and 10x + 20y ≥ 40

Also, x ≥ 0, y ≥ 0

The cost of x units of F1 and y units of F2 is x + 2y
Since the cost is to be minimized, therefore the LPP is