Optimise the problem question

I have the following question to attempt:

A company manufactures two types of products, ‘A’ and ‘B’. The production cost associated with these products are £200 and £150 respectively. Selling each unit of product ‘A’ brings £20 and product ‘B’, £15 of profit. The monthly demand for both products combined is 500 units. The production budget is limited to £50,000. How many units of each product should be made to gain maximum profit?

Right now I have my function as:

How do i know what my starting point (x0) is?

and should I have a value for Aeq and Beq?

Thanks in advance for any help provided.

[Walter Roberson]

You can construct an UB according to max() of the budget divided by the two unit costs.

[Walter Roberson]

It does not matter how many units of each you make as long as the total cost is the budget. The profit is 10% no matter which you make, so any combination that adds up to 50000 will get you 5000 profit.

(though this does not directly take into account the 500 unit requirement)

Answer 1

[Matt J]

You should be using linprog to solve this problem. It does not require a starting point x0. I don't see any need for equality constraints, but I don't see where you have satisfied this constraint: "The monthly demand for both products combined is 500 units"

[Teanna Mills]

We have only been taught fmincon within the module and we have been advised to use it again - can it not be applied in this case?

I understand that I have not satisfied the constraint of the 500 units - would this be within the Aeq section?

[Matt J]

You can use it. It's just not the most efficient tool. It doesn't know, and so cannot take advantage of the fact, that the problem is linear. In any case, the problem is convex, so if you are going to use fmincon, any x0 should work equally well - there are no local minima to get stuck in.

I understand that I have not satisfied the constraint of the 500 units - would this be within the Aeq section?

To me it sounds like the constraint requires at least 500 units to be produced.

[Walter Roberson]

You could produce more than 500 but you cannot sell, them, so the profit would go down. Therefore in practice the limit is 500 exactly.

[Matt J]

Ah, and not only that, but now I notice that x(1) and x(2) represent the amounts that are actually sold, not the amounts produced. So, in fact you would have the constraint

x(1)+x(2)<=500

In this case, since there are no other upper constraints on x(i) then, as Walter says, the constraint will be satisfied with equality at the solution

x(1)+x(2)=500

More generally, though, if you had upper bounds on the x(i) or other constraints, you could not assume that the full demand could be met and you would have to impose inequality.

[Teanna Mills]

would that constraint then require to be applied as Aeq as it is an equality?

[Matt J]

There is no requirement that it be applied with equality. As discussed above, the optimizer will force equality to be reached regardless of whether you specify the constraint as an equality or inequality. But if you wished to specify equality, Aeq is what you would use.