1. ## Linear Programming Problem

Hello!

I have a problem for you. Here is a description. I have raw material (cocoa beans) from which I get 1 ton of cocoa liquor per hour. The price of cocoa liquor is let say 3.300 USD/ton. From 1 ton of cocoa liquor I can produce 0.35 ton of cocoa butter and 0.48 ton of cocoa powder. The price of butter is let say 5.000 USD/ton and the price of powder is lets say 1.200 USD/ton.

cocoa beans->
cocoa liquor (1 ton, 3300 USD/ton)->cocoa butter(0.35 ton, 5000 USD/ton)
->cocoa powder(0.48 ton, 1200 USD/ton)

The ratio between cocoa butter and cocoa powder is fixed. Cocoa liquor can be sold as independent product as it is or it can be further processed into cocoa butter and cocoa powder. The question now is, how much cocoa liquor should I sell as independant product and how much cocoa liquor should I further process into cocoa butter and cocoa powder to maximize the profit? How should I deal with this? Please help!

2. 1 ton of liquor = $3.30 1 ton of liquor converted to powder & butter (.35*5+.48*1.2) =$2.326

So 1 ton of cocoa liquor can either be sold as is for $3.30,or processed (which will cost money, even though that is not given in the problem) and sold for$2.326

3.3 > 2.326
So I would sell it all as liquor, and convert none of it.

3. Thank you for your answer. That is one point of view. But what if there is a demand on the market for butter and powder and I'd have to process some amount of liquor to butter and powder? But probably the logic would be to process as little as possible since it doesn't pay off to process liquor any further, but what if I have to?

4. Originally Posted by direktor
Thank you for your answer. That is one point of view. But what if there is a demand on the market for butter and powder and I'd have to process some amount of liquor to butter and powder? But probably the logic would be to process as little as possible since it doesn't pay off to process liquor any further, but what if I have to?
Well, I suppose this question changes if you go from a math problem to a real life problem.

If I were considering this in real life, you have many variables. For example, how much can you actually produce? How much can you actually sell?
Ie if you can produce 100,000 tons a day, and sell all of it as liquor, then there is no reason to produce any butter/powder. Realistically, though, the more you can produce, the greater supply is, and this changes the balance of supply and demand. Is there a formula for how much cocoa beans cost as a function of how much you buy?.

If you produce 100,000 tons, you might have to lower your price to sell it all, but if you produce 10,000 tons, you might be able to raise your price and still sell it all. If you come up with a mathematical formula that seems to adequately indicate how much you can sell depending on price, then you could look at what the optimal quantity to sell as liquor and the optimal quantity to process to butter/powder is.

But these things take more information. ie, how much can you produce, how much can you sell (or if you can sell all of it, is there some sort of formula that will reflect what price you need to charge in order to sell a given amount), what is the cost to convert to powder and butter? (this will include not only the cost of processing, but of the different type of packaging, the cost of creating new packaging, the shipping costs will be different, etc), how much powder you can sell and/or a formula for determining how much you can sell as a function of price.

Basically, if you want to use this in real life, you need a lot more information than what is given. But based only on what is given, there is no reason to even look at powder/butter, because every ton you convert into powder/butter, will be sold for less than if you had just sold it as liquor.

In real life, there are other implied considerations too, for example, if you produce butter, powder, and liquor, then your business has several products to offer, this could be beneficial in and of itself, as someone might be purchasing powder, and say "you know, I could use some liquor too" and because they are already purchasing powder from you, they also purchase liquor from you. Basically, multiple products help market eachother (think how you go to wallmart to buy a shirt, but end up picking up a belt and socks too).

Anyway, I'm not entirely sure the scope of your question, if it's a simple school question, then I would say buy all liquor. If it is a real life question, I would say you need to do some research and consolidate your information before you make any decisions.

5. Originally Posted by direktor
Thank you for your answer. That is one point of view. But what if there is a demand on the market for butter and powder and I'd have to process some amount of liquor to butter and powder? But probably the logic would be to process as little as possible since it doesn't pay off to process liquor any further, but what if I have to?
Even to make this a linear programming problem you need more information
on the constraints. We can guess an objective but without constraints this
is trivial.

RonL

6. Originally Posted by angel.white
as someone might be purchasing powder, and say "you know, I could use some liquor too" and because they are already purchasing powder from you, they also purchase liquor from you. Basically, multiple products help market eachother (think how you go to wallmart to buy a shirt, but end up picking up a belt and socks too).
Exactly. It's a real life problem. Let's suppose I'll be able to sell everything I produce. I don't have any formulas of supply and demand. The constraints would be:

The machines are able to process 1 ton of beans per hour, which gives me 0.83 ton of liquor that can be further processed into butter (0.35 ton per hour) and powder (0.48 ton per hour). Factory can operate 6.000 hours per year which gives me 6.000 tons of cocoa beans, 4.920 tons of liquor, 2.100 tons of butter and 2.880 tons of powder per year. Prices are as mentioned:

butter = 5.000 USD/ton
price = 1.200 USD/ton
liquor = 3.300 USD/ton

Packaging included. This is the final selling price. I don't know exactly the partial cost of further processing liquor to butter and powder, but it's a good question. I'll have to look into it. Do you need and other info or is that enough?

7. The price of beans is about 1.500 USD/ton but that's probably irrelevant.