how Determine the maximum values of C.

Question

how Determine the maximum values of C.

my try is that :

To graph the last two bounding lines, I'll want to put the equations into slope–intercept form.

The bounding line corresponding to the 3rd constraint inequality becomes:

2y = 8 + 4x

y = 4 + 2x

And the bounding line corresponding to the 4th constraint inequality becomes:

y = 4 – 2x

I need to know how get From the graph the vertices of the region to get the the maximum values of C.

Answer 1

When working on Linear programming problems, we use a consistent approach by attempting to:

• Evaluate the objective function $C$ for each labeled point in the feasible region
• Pick points on the region endpoints and points inside the region.
• At which labeled point does the maximum value of $C$ occur? At which labeled point does the minimum value of $C$ occur?
• What are the maximum and minimum values of $C$ on the entire feasible region?
• Try other constraints using points in the region to see if you can find values $x$ of $C$ that are greater or lesser than those you $y$ found.

Lets try this approach on your problem. If we draw the first two constraints ($0 \le x \le 6, 0 \le y \le 6$), we have the feasible region:

If we then add the last two constraints, the feasible region (the top left cutout is from $y \le 2x + 4$ and the bottom left cutout is from $y \ge -2x + 4$) is:

Now, we want to maximize:

$$C(x, y) = 3x + y$$

We can test each of the vertices as:

• $C(2, 0) = 6$
• $C(6, 0) = 18$
• $C(0, 4) = 4$
• $C(1, 6) = 9$
• $C(6, 6) = 24$
• Per the approach described above, we can also test points inside the region and see that they will not provide a maximum.

It is clear which is the maximum.

We could have also noted that the function $3x + y$ is increasing for positive $x$ and $y$ and from the feasible region plot, the max is given by $(x, y) = (6, 6)$. The same technique can be used to determine the minimum point.