Linear Programming Constraints

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TLamb

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I'm not sure that I've identified all of the constraints. I can't seem to determine what the 2-variable equation should be (ie, x + y = z). Here's the problem statement:
To make one package of all-beef hot dogs, a manufacturer uses 1lb of beef. To make one package of regular hot dogs, the manufacturer uses 1/2 lb of each beef and pork. The profit on the all-beef hot dogs is $0.45 per pack and the profit on the regular hot dogs is $0.35 per pack. There are 300 lbs of beef and 200 lbs of pork. List the constraints. Determine the objective function.

Trying to maximize profit - P = 45x + 35y.
So far, the constraints I've got are:

100 <= x <= 300
0 <= y <= 400

x is the pkgs of beef hot dogs, y is the pkgs of regular hot dogs.
 
TLamb said:
I can't seem to determine what the 2-variable equation should be (ie, x + y = z).

Your example equation appears to contain three variables: x, y, and z.

You must be trying to say something else.


100 <= x <= 300 This needs adjusting.


0 <= y <= 400 This looks okay.


x is the pkgs of beef hot dogs, y is the pkgs of regular hot dogs. We can improve this wording.

x = the number of all-beef packages produced

y = the number of regular packages produced


Trying to maximize profit - P = 45x + 35y This is not the correct expression for profit.

The dollar amounts are wrong; use the same dollar amounts provided in the exercise.

NOTE: When typing equations, it's not a good idea to preface them with a hyphen.

Some people might read - P to mean the opposite of P. It's better to type each equation on its own line without punctuation marks or English phrases.
Click to expand...

One constraint is the volume of beef; there is no more than 300 pounds available.

Each all-beef package accounts for 1 pound of beef:

(1)(x) = pounds of beef used in x packages of all-beef product

Each regular package accounts for 1/2 pound of beef:

(1/2)(y) = pounds of beef used in y packages of regular product.

Therefore, the total amount of beef used in both products is: x + 1/2 y

x + 1/2 y < 300

It appears that you already arrived at the inequality representing the pork constraint.

0x + 1/2 y < 200

In linear-programming exercises, I don't often see compound inequalities written.

I would have stated four separate inequalities:

x > 0

y > 0

x + 1/2 y < 300

1/2 y < 200

These four inequalities form a system, and the graph of this system's solution set is a four-sided shaded region, where the maximum profit is realized at one of the four vertex points.

This exercise does not ask for solution values for x and y (i.e., the specific pair of values which maximize profit), so you do not need to graph the four lines.

Perhaps, your class is taking a different approach.

If you're supposed to write the bounds on x (like you did for y), then you'll need to find the x-intercept of the following line because that's the largest x without y going negative.

x + 1/2 y = 300

Thank you for showing your work.

Cheers ~ Mark 8-)

MY EDITS: Deleted comments pertaining to finding an actual solution; I initially misread the given instructions. Also, I tried to reduce the verbosity of my prose.
 
The objective function is to maximize profit.

\(\displaystyle z=.45x+.35y\)

The constraints are

\(\displaystyle x+\frac{1}{2}y\leq 300\)

\(\displaystyle \frac{1}{2}y\leq 200\)

I ran this through Excel Solver and got a solution. Is that what you're using?.
 
Thank you so much for the quick response.
If x is the number of all-beef packages, and y is the number of regular packages, how to you arrive at x + 1/2y <= 300?
 
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