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Question 460454: I am really struggling with these homework problems and could use some assistance in figuring out how to solve them.
3. Suppose we are to maximize the objective function 4x + 3y over a feasible set that has vertices (1, 2), (5, 0), (8, 3), and (6, 4). The maximum objective function value is ______. I answered (8,3) but it was incorrect. How do I solve this?
4. The XYZ Corporation plans to open three different types of fast-food restaurants. Each type of A restaurant requires an initial cash outlay of $600,000, needs 15 employees, and is expected to make an annual profit of $45,000. Each type B restaurant requires an initial cash outlay of $400,000, needs 9 employees, and is expected to make an annual profit of $35,000. Each type C restaurant requires an initial cash outlay of $300,000, needs 5 employees, and is expected to make an annual profit of
$25,000. The XYZ Corporation has $48,000,000 available for initial outlays, does not want to hire more than 1000 new employees, and would like to open at most 70 restaurants. How many restaurants
of each type should the XYZ Corporation open in order to maximize the expected annual profit? Let x = number of type A restaurants, y = number of type B restaurants, and z = number of type C restaurants.
I said 45000x+35000y+25000z = 70 subject to
{x+y+z<=45000
600x+400y+300z<=35000
15x+9y+5z<=25000
x,y,z>=0
I got it wrong as well. Please show me how to do these two problems.
Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website! 3. Suppose we are to maximize the objective function 4x + 3y over a feasible set that has vertices (1, 2), (5, 0), (8, 3), and (6, 4). The maximum objective function value is ______. I answered (8,3) but it was incorrect. How do I solve this?
You didn't read the question carefully. You were asked
to give the maximum value of 41, not the coordinates of
the vertex at which this maximum occurred. You were right
that (8,3) is the vertex at which the maximum value occurs,
but that's not what the problem asked for.
When you evaluate 4x + 3y at each of the
vertices:
x y | 4x + 3y
---------------------------
1 2 | 4(1)+3(2) = 4+6 = 10
5 0 | 4(5)+3(0) = 20+0 = 20
8 3 | 4(8)+3(3) = 32+9 = 41
6 4 | 4(6)+3(4) = 24+12 = 36
So the correct answer is 41, not (8,3)
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4. The XYZ Corporation plans to open three different types of fast-food restaurants. Each type of A restaurant requires an initial cash outlay of $600,000, needs 15 employees, and is expected to make an annual profit of $45,000. Each type B restaurant requires an initial cash outlay of $400,000, needs 9 employees, and is expected to make an annual profit of $35,000. Each type C restaurant requires an initial cash outlay of $300,000, needs 5 employees, and is expected to make an annual profit of
$25,000. The XYZ Corporation has $48,000,000 available for initial outlays, does not want to hire more than 1000 new employees, and would like to open at most 70 restaurants. How many restaurants
of each type should the XYZ Corporation open in order to maximize the expected annual profit? Let x = number of type A restaurants, y = number of type B restaurants, and z = number of type C restaurants.
We get the cash outlay statements together:
Each type A restaurant requires an initial cash outlay of $600,000
Each type B restaurant requires an initial cash outlay of $400,000
Each type C restaurant requires an initial cash outlay of $300,000
The XYZ Corporation has $48,000,000 available for initial outlays
Total of cash outlays = 600000x + 400000y + 300000z ≦ 48000000
---------------------------------------------------------------
We get the employees statements together:
Each type A restaurant...needs 15 employees
Each type B restaurant...needs 9 employees
Each type C restaurant...needs 5 employees
The XYZ Corporation...does not want to hire more than 1000 new employees
Total number of employees = 15x + 9y + 5z ≦ 1000
---------------------------------------------------------------
We get the expected profit statements together:
Each type A restaurant...is expected to make an annual profit of $45,000
Each type B restaurant...is expected to make an annual profit of $35,000.
Each type C restaurant...is expected to make an annual profit of $25,000.
Total expected profit = 45000x+35000y+25000z = P
This is what we are to maximize. So the problem is to
maximize P = 45000x+35000y+25000z
Subject to the constraints:
600000x + 400000y + 300000z ≦ 48000000
15x + 9y + 5z ≦ 1000
x,y,z ≧ 0
That may be what you're looking for without solving.
I'll start to solve it though.
To solve it we can simplify it by letting the profit be
in thousands of dollars rather than in dollars, and we can
divide the first inequality through by 100000
Then our
we have:
maximize P = 45x+35y+25z subject to the constraints:
6x + 4y + 3z ≦ 480
15x + 9y + 5z ≦ 1000
x,y,z ≧ 0
So we put in two slack variables and make the equations:
6x + 4y + 3z + s1 = 480
15x + 9y + 5z + s2 = 1000
Then we get 0 on the right of objective function:
P = 45x + 35y + 25z
-45x - 36y - 25z + P = 0
And we have this system of equations:
6x + 4y + 3z + s1 = 480
15x + 9y + 5z + s2 = 1000
-45x - 36y - 25z + P = 0
Now we make this matrix:
⎾ 6 4 3 | 1 0 | 0 | 480⏋
⎢ 15 9 5 | 0 1 | 0 | 1000⎢
⎢—————————————————————————————⎢
⎿-45 -36 -25 | 0 0 | 1 | 0⏌
Do you want me to finish the Simplex method here?
It get's pretty messy. Maybe you were just supposed
to set it up. If you were to solve it, post again
or tell me in a thank-you note.
Edwin
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