Would it help if I provided you with a similar question and its solution?
If so, let's say our problem is:
Solution:Let the sick person take x units and y units of food A and B respectively that were taken in the diet.
Since, per unit of food A costs at Tk. 4 and that of food B costs Tk. 3.
Therefore, x units of food A costs Tk. 4x and y units of food B costs Tk. 3y.
Total cost = Tk. (4x + 3y)
Let Z denote the total cost
Then, Z = 4x + 3y
If one unit of A contains 200 units of vitamins and one unit of food B contains 100 units of vitamins.
Thus, x units of food A and y units of food B contains 200x + 100y units of vitamins.
But a diet for a sick person must contain at least 4000 units of vitamins.
∴ 200x+100y≥4000
If one unit of A contains 1 unit of mineral and one unit of food B contains 2 units of mineral.
Thus, x units of food A and y units of food B contains x + 2y units of mineral.
But a diet for a sick person must contain at least 50 units of minerals.
∴ x+2y ≥ 50
If one unit of A contains 40 calories and one unit of food B contains 40 calories.
Thus, x units of food A and y units of food B contains 40x + 40y units of calories.
But a diet for a sick person must contain at least 1400 calories.
∴40x+40y≥1400
Finally, the quantities of food A and food B are non-negative values.
So, x, y ≥ 0
Hence, the required LPP is as follows:
Min Z = 4x + 3y
subject to
200x+100y≥4000 x+2y ≥50 40x + 40y ≥ 1400 x, y≥0
First, we will convert the given inequalities into equations, we obtain the following equations:
200x + 100y = 4000, x +2y = 50, 40x + 40y =1400, x = 0 and y = 0
Region represented by 200x + 100y ≥ 4000:
The line 200x + 100y = 4000 meets the coordinate axes at (20, 0) and (0,40) respectively. By joining these points we obtain the line
200x + 100y = 4000. Clearly (0,0) does not satisfies the inequality 200x + 100y ≥ 4000. So, the region in x-y plane which does not contain the origin represents the solution set of the inequality 200x + 100y ≥ 4000.
Region represented by x +2y ≥ 50:
The line x +2y = 50 meets the coordinate axes at (50, 0) and (0, 25) respectively. By joining these points we obtain the line
x +2y = 50.Clearly (0,0) does not satisfies the x +2y ≥ 50. So, the region which does not contains the origin represents the solution set of the inequality x +2y ≥ 50.
Region represented by 40x + 40y ≥ 1400:
The line 40x + 40y = 1400 meets the coordinate axes at (5, 0) and (0, 35) respectively. By joining these points we obtain the line
40x + 40y = 1400.Clearly (0,0) does not satisfies the inequality 40x + 40y ≥ 1400. So, the region which does not contains the origin represents the solution set of the inequality 40x + 40y ≥ 1400.
Region represented by x ≥ 0 and y ≥ 0:
Since, every point in the first quadrant satisfies these inequalities. So, the first quadrant is the region represented by the inequalities x ≥ 0, and y ≥ 0.
The feasible region determined by the system of constraints 200x + 100y ≥ 4000, x +2y ≥ 50, 40x + 40y ≥ 1400, x ≥ 0, and y ≥ 0 are as follows.
The corner points of the feasible region are A(0, 40), B(5, 30), C(20, 15) and D(50, 0)
The value of the objective function at these points are given by the following table
Please reply back if you're still confused.