Optimizing Costs: A Scientific Approach to Minimizing Company Expenses

[space-time]

Homework Statement

The demand for some bikes that a company sells is 10000 bikes per year, and they sell at a uniform rate throughout the year. The cost of one shipment of bikes is $10000 and the cost of storing each bike is $200 per year. If the company orders too many bikes at once, the storage cost increases, and if they order too often, the shipment cost increases. How large should each order be, and how often should the orders be placed to minimize the ordering and storage costs?

Relevant Equations

N/A (or at least I don't know of any specific equations that are particularly relevant here)

I first tried to set up an expression for the total amount of money that the company would spend in a year. That would be:

p(x, y) = 10000x + 200xy

where:
p(x, y) = the amount of money (dollars) that the company would spend in a year.
x = the number of shipments ordered per year.
y = the number of bikes per shipment

Now, I was thinking that this would be an optimization problem where I would have to find a local minimum for the function p. However,

p_x = 10000 + 200y
p_y = 200x

These partial derivatives only ever equal 0 at x = 0 and y = -50.

That presents quite the problem, seeing as how the company would certainly need to order more than 0 shipments per year, and there cannot be -50 bikes per shipment.

Hence, I am stuck. Is my approach all wrong to begin with?

 

[BvU]

Now, I was thinking that this would be an optimization problem where I would have to find a local minimum for the function p

Mimimizing the cost is NOT what a company is trying to do. Maximizing earnings is. So you need a bit more. I have no idea whether you have enough information to answer the question, but the next thing I would do in your case is: make a few assumptions and hope they drop out of the equation ...

[edit] A second look comparing your effort with the given info: You don't use the second 10000, the amount of bikes sold per year !$\$

 

[Office_Shredder]

Mimimizing the cost is NOT what a company is trying to do. Maximizing earnings is. So you need a bit more. I have no idea whether you have enough information to answer the question, but the next thing I would do in your case is: make a few assumptions and hope they drop out of the equation ...

$\$

The revenue here is fixed - they sell 10,000 bikes, at whatever cost they sell them for. The only thing to do here is minimize the cost of the shipment.

The piece you are missing is this is a constrained optimization. xy=10,000 because that's how many bikes they ship in a year.

 

[BvU]

My self-esteem now shredded by OS , I nevertheless venture one more comment:

Your cost function needs reconsidering: if they order only one shipment per year, the cost is not 200 y



$\$

 

[space-time]

The revenue here is fixed - they sell 10,000 bikes, at whatever cost they sell them for. The only thing to do here is minimize the cost of the shipment.

The piece you are missing is this is a constrained optimization. xy=10,000 because that's how many bikes they ship in a year.

If I apply the xy = 10000, then the function becomes
p(x) = 10000x + 2000000

The problem is that this is a linear function of x and has no minimum.

If I break it down into shipment costs and storage costs respectively, it would be like this:

shipment cost = 10000x
storage costs = 200xy = 2000000 (since xy = 10000)

They're going to spend $2,000,000 on storage either way from the looks of it.

If that's the case, would it be accurate to say that they should just order 1 big shipment of 10000 bikes for the whole year? After all, if they're going to spend 2 million on storage regardless, then the only way to minimize costs now is to keep the number of shipments to a minimum (which would just be 1 shipment).

 

[BvU]

The 2 million is not right: If they order two times per year, they only have to store how many bikes (on average) ?

$\$

 

[space-time]

The 2 million is not right: If they order two times per year, they only have to store how many bikes (on average) ?

$\$

That would be 5000 bikes right? (Because they need 10000 for the whole year)

 

[Office_Shredder]

That would be 5000 bikes right? (Because they need 10000 for the whole year)

How long does it take to sell the bikes they get each shipment? They ship 5,000 bikes, and then they sell them.

 

[space-time]

How long does it take to sell the bikes they get each shipment? They ship 5,000 bikes, and then they sell them.

The problem doesn't tell us how long they take to sell the bikes. It just says that they sell at a uniform rate throughout the year.

 

[Office_Shredder]

The problem doesn't tell us how long they take to sell the bikes. It just says that they sell at a uniform rate throughout the year.

They sell 10,000 bikes uniformly throughout the year, so how long does it take to sell half as many bikes?

 

[space-time]

They sell 10,000 bikes uniformly throughout the year, so how long does it take to sell half as many bikes?

Half of a year

 

[BvU]

That would be 5000 bikes right? (Because they need 10000 for the whole year)

No!

 

[Office_Shredder]

Half of a year

So when they get the shipment, they have to store 5,000 bikes. They then start selling them, and six months later, they have no bikes in storage. How long on average does each bike sit in storage? Remember, one of them got sold immediately, and one of them takes six months to sell, so the answer should be somewhere in between those...

 

[space-time]

So when they get the shipment, they have to store 5,000 bikes. They then start selling them, and six months later, they have no bikes in storage. How long on average does each bike sit in storage? Remember, one of them got sold immediately, and one of them takes six months to sell, so the answer should be somewhere in between those...

Each bike would stay in storage for an average of about 3 months?

 

[Office_Shredder]

So how much do they spend in storage costs? It's probably time for you to take a little initiative here, try to write down a new formula for how much they actually spend in costs.