Hello I am working on an optimization problem and I want to make sure if I am doing it right. I will post the question and my attempt at the solution below. The best response will receive a medal:

Question

anonymous · Answer

Consider the production scheduling problem of the perfume Polly named after a
famous celebrity. The manufacturer of the perfume must plan production for the first
four months of the year and anticipates a demand of 4000, 5000, 6000, and 4500
gallons in January, February, March, and April, respectively. At the beginning of
the year the company has an inventory of 2000 gallons. The company is planning
on issuing a new and improved perfume called Pollygone in May, so that all Polly
produced must be sold by the end of April. Assume that the production cost for
January and February is $5 per gallon and this will rise to $5.5 per gallon in March
and April. The company can hold any amount produced in a certain month over to
the next month at an inventory cost of $1 per unit. Formulate a linear optimization
model that will minimize the costs incurred in meeting the demand for Polly in the
period January through April. Assume for simplicity that any amount produced in a
given month may be used to fulfill demand for that month.

My attempt at the solution is.
minimize: $$5x_1$$$$+5x_2+5.5x_3+5.5x_4$$
constraints $$t \le 4 months$$
$$5x_1 \le 4000$$
$$5x_2 \le 5000$$
$$5.5x_3 \le 6000$$
$$5.5x_4 \le 4500$$
$$x_1, x_2, x_3, x_4 \ge 0$$

anonymous · Answer

I'm not sure how the 2000 gallons of initial inventory plays a role or the $1 per unit cost carried over

tkhunny · Answer

What are the units on $5x_{1}$?
What are the units on $4000$?

anonymous · Answer

I see $$5x_1$$ would be in dollars and 4000 is in gallons. My actual constraints should be $$x_1 \le 4000$$ because that is comparing gallons to gallons. The same would be of the others without 5 or 5.5 in front of the x variable. However I'm still not sure about what I had mentioned before. Or was I supposed to incorporate it within my constraints? Thank you!

tkhunny · Answer

Now, where are 2000 you started with?

anonymous · Answer

Since the 2000 can be used to fulfill the demand for the month of January that means I would not need to be worried about it so it is $$x_1 \le 2000$$?

anonymous · Answer

and the others remain the same.

anonymous · Answer

However I still don't know about the $1/ unit I know I am going to need to multiply it somewhere though

tkhunny · Answer

One thing at a time.  We'll get there.

$x_{1} + 2000 \le 4000$

anonymous · Answer

Ok sorry for asking multiple questions at a time. Is it more convenient to right it that way as a way of showing steps?
I'll think about it in the meantime.

tkhunny · Answer

Tough question...  How many do you NOT sell in January?

anonymous · Answer

I think my reasoning behind this answer will be wrong, but I believe I would not sell 2000 because even though 4000 will be in demand 2000 of those will be from the previous inventory and as a result 2000 will probably carry on to the other months(so I would add 2000 to those months too)?

tkhunny · Answer

Too smooth.  People don't behave that way.  Define a Slack Variable.

$x_{1} + 2000 \le 4000$

is EXACTLY the same as 

$x_{1} + 2000 + x_{5} = 4000$  <== Note the Equal Sign.  We've accounted for ALL of it.

$x_{1}$ is the January Production
$x_{5}$ is the Inventory Holdover for February.

anonymous · Answer

So by using a slack variable, once everything is accounted for I no longer need to worry about introducing slack variables into the others since this took care of all of them? Or did this only account for all of it in January?

tkhunny · Answer

Nope.  Different one every month.  Add them to your cost equation.

anonymous · Answer

so is my cost equation now: $$5x_1+5x_2+5.5x_3+5.5x_4+x_5+x_6+x_7+x_8$$?

anonymous · Answer

Do I need to add 200 to the others also?

anonymous · Answer

2000*

tkhunny · Answer

You don't get one in April.  It's in the problem statement.  It must be equal to 4500 without a slack variable.

tkhunny · Answer

January

$2000 + x_{1} + x_{5} = 4000$

February

$x_{5} + x_{2} + x_{6} = 5000$

March

$x_{6} + x_{3} + x_{7} = 6000$

You do April.

anonymous · Answer

For April I got $$x_7+x_4 \le 4500$$ since $$x_7$$ is still the inventory from the previous month that got carried over?
And my knew cost equation is: $$5x_1+5x_2+5.5x_3+5.5x_4+x_5+x_6+x_7$$ ?

tkhunny · Answer

Not quite.  "= 4500"  You don't get any slack on this one.  In other words, $x_{8} = 0$ is another constraint.  Or, just leave it out.

Now, do a little algebra on January, and you'll be off to find a solution.

anonymous · Answer

So instead it would be strictly less than 4500?:
$$x_7+x_4 \le 4500$$
Are my slack variables the ones that are being multiplied by the $1/unit(this is what I believe since it is the inventory being carried over)? Or do I do the algebra on January and solve for the variables then use it?

tkhunny · Answer

I gave you January.  Subtract 2000 from both sides and it will be done.

You don't seem to have the spirit of April.  You must sell EVERYTHING.  There can be nothing left.  It is strict EQUALITY.  No inequality at all.

anonymous · Answer

Sorry I meant to write $$x_7+x_4=4500$$ since as you said everything is being sold?. After I subtract 2000 from both sides that means I will be left off with $$x_1+x_5=2000$$. I'm thinking cost equation is correct and also my reasoning of $1/unit? I apologize if I am going ahead of questions without finishing the current one again.

tkhunny · Answer

No worries.  I think we laid out the whole thing.  Excellent work.

You may not have encountered a "carry-over" sort of constraint, before.  Now, you have.

anonymous · Answer

Thank you very much! Yes today was the first day of class, so I was not familiar with the "carry-over" constraint. Thank you for guiding me!

tkhunny · Answer

First day?  You are doing VERY well!  Go forward without fear.  :-)