Refer to Exercise 12. What will be the minimum cost?

Referring to the exercise 12, we get following data

Let the firm has x number of large vans and y number of small vans. We make the following table from the given data:

Thus according to the table, the cost becomes, Z=400x+200y

Now, we have to minimize the cost, i.e., minimize Z=400x+200y

The constraints so obtained, i.e., subject to the constraints,

200x+80y≥ 1200

Now will divide throughout by 40, we get

⇒ 5x+2y≥ 30…………..(i)

And 400x+200y≤3000

Now will divide throughout by 200, we get

⇒ 2x+y≤ 15…………..(ii)

Also given the number of large vans cannot exceed the number of small vans

⇒ x≤ y……………..(iii)

And x≥0, y≥0 [non-negative constraint]

So, minimize cost we have to minimize, Z=400x+200y, subject to

5x+2y≥ 30

2x+y≤ 15

x≤ y

x≥0, y≥0

Now let us convert the given inequalities into equation.

We obtain the following equation

5x+2y≥ 30

⇒ 5x+2y=30

2x+y≤ 15

⇒ 2x+y=15

x≤ y

⇒ x=y

x ≥ 0

⇒ x=0

y ≥ 0

⇒ y=0

The region represented by 5x+2y≥ 30:

The line 5x+2y=30 meets the coordinate axes (6,0) and (0,15) respectively. We will join these points to obtain the line 5x+2y=30. It is clear that (0,0) does not satisfy the inequation 5x+2y≥ 30. So the region that does not contain the origin represents the solution set of the inequation 5x+2y≥ 30

The region represented by 2x+y≤ 15:

The line 2x+y=15 meets the coordinate axes (7.5,0) and (0,15) respectively. We will join these points to obtain the line 2x+y=15. It is clear that (0,0) satisfies the inequation 2x+y≤ 15. So the region that contain the origin represents the solution set of the inequation 2x+y≤ 15

The region represented by x≤y:

The line x=y is a line that passes through the origin and doesn’t touch any coordinate axes at any other point except (0,0). We will join these points to obtain the line x=y. It is clear that (0,0) satisfies the inequation x≤y. So the region that contain the origin represents the solution set of the inequation x≤y

Region represented by x≥0 and y≥0 is first quadrant, since every point in the first quadrant satisfies these inequations

The graph of these equations is given.

The shaded region ABC represents the feasible region is bounded, and minimum value will occur at a corner point of the feasible region.

Corner Points are , B (0, 15) and C(5, 5)

Now we will substitute these values in Z at each of these corner points, we get

So from the above table the minimum value of Z is at point ,

Hence, the minimum cost of the firm is Rs. 2571.43.