9. A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1 costs Rs 4 per unit food and F2 costs Rs 6 per unit. One unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.
Let diet contain x unit of food F1 and y unit of foof F2 .Thus, .
The given information can be represented in table as :
Vitamin | minerals | cost per unit | |
foof F1 | 3 | 4 | 4 |
food F2 | 6 | 3 | 6 |
80 | 100 |
Cost of food F1 is Rs 4 per unit and Cost of food F2 is Rs 6 per unit
Therefore, constraint are
The feasible region determined by constraints is as follows:
We can see feseable region is unbounded.
The corner points of feasible region are
The value of Z at corner points is as shown :
Corner points | ||
106.67 | ||
104 | minimum | |
200 | maximum | |
Feasible region is unbounded , therefore 104 may or may not be minimum value of Z .
For this we draw or and check whether resulting half plane has point in common with feasible region or not.
We can see feasible region has no common point with .
Hence , Z has minimum value 104 .