Question

Find the maximum value of the objective function and the values of x and y for which it occurs. F= 2x+y3x+5y ≤45 x ≥0 and y ≥02x+4y ≤32The maximum value of the objective function is ______.

Answer 1

The objective function is:

`F=2x+y`

We need to find the shaded region where 4 inequalities overlap, then we need to graph the given inequalities:

`\begin{gathered} 3x+5y\leq45 \\ 5y\leq-3x+45 \\ y\leq(-3)/(5)x+(45)/(5) \\ y\leq-(3)/(5)x+9 \end{gathered}`

And the other one:

`\begin{gathered} 2x+4y\leq32 \\ 4y\leq-2x+32 \\ y\leq(-2x)/(4)+(32)/(4) \\ y\leq-(1)/(2)x+8 \end{gathered}`

And x>=0, y>=0

The graph is:

The coordinates of the shaded region are then:

(0,0), (0,8), (10,3) and (15,0)

To obtain the maximum value, let's evaluate the objective function in all the coordinates:

`\begin{gathered} F(0,0)=2*0+0=0+0=0 \\ F(0,8)=2*0+8=0+8=8 \\ F(10,3)=2*10+3=20+3=23 \\ F(15,0)=2*15+0=30+0=30 \end{gathered}`

Then, the maximum value is the largest value obtained, then it's 30 and it occurs at x=15 and y=0.