How do you set up this lp problem? I'm not sure how to set up the constraints.

[nueton5000]

A company produces three products A1, A2, A3 by mixing three ingredients B1, B2, B3. The selling price for A1, A2 and A3 is $13, 14 and 16 $/kg, respec- tively, and at most 75,80 and 90 kg of each can be sold daily. The cost of B1,B2,B3 is 7, 2 and 4 $/kg and the daily supply is at most 40, 95, and 65 kg, respectively. In addition, there are the following technological constraints:
A1 must contain exactly 10% of B1, at least 30% of B2, and at most 50% of B3;
A2 must contain exactly 30% of B2, at least 20% of B3, and at most 10% of B1.
A3 must contain exactly 15% of B3, at least 15% of B2, and at most 15% of B1;

*Determine the mixing scheme which will maximize the profit

The Attempt at a Solution

I think I need to maximize 13a1 + 14a2 + 16a - 7b1 -2b2 -4b3 but I am not sure how the constraints for the A(1-3) products would go.

 

[Ray Vickson]

A company produces three products A1, A2, A3 by mixing three ingredients B1, B2, B3. The selling price for A1, A2 and A3 is $13, 14 and 16 $/kg, respec- tively, and at most 75,80 and 90 kg of each can be sold daily. The cost of B1,B2,B3 is 7, 2 and 4 $/kg and the daily supply is at most 40, 95, and 65 kg, respectively. In addition, there are the following technological constraints:
A1 must contain exactly 10% of B1, at least 30% of B2, and at most 50% of B3;
A2 must contain exactly 30% of B2, at least 20% of B3, and at most 10% of B1.
A3 must contain exactly 15% of B3, at least 15% of B2, and at most 15% of B1;

*Determine the mixing scheme which will maximize the profit

The Attempt at a Solution

I think I need to maximize 13a1 + 14a2 + 16a - 7b1 -2b2 -4b3 but I am not sure how the constraints for the A(1-3) products would go.

Suppose we let the letters A1,A2 and A3 do double duty: let A1 = daily production of product A1 (in kg), etc. So, if we produce A1 kg of product A1, how many kg of B1 do we use (assuming the percentage figures are by weight)? It is a bit trickier for B2, since the amount of B2 in A1 is bounded, but not given as an exact percentage. So, the daily amount of B2 in A1 is another _variable_---call it B2A1 for example. What, if any, constraint links B2A1 and A1?

By introducing a bunch of extra variables, the problem becomes easy to model.

RGV

 

[nueton5000]

Thank you but I'm still a little confused. So would the constraint be b2a1/a1 >= .3 ?

 

[Ray Vickson]

Thank you but I'm still a little confused. So would the constraint be b2a1/a1 >= .3 ?

Well, isn't that what one of the restrictions actually says? However, you cannot write it like that because you would be taking a *ratio* of variables, which is a NONLINEAR expression (and not only that, you would be dividing by zero if you happen to look at a policy with A1 = 0). You need to put the restriction in the form of a LINEAR constraint. There are standard ways of doing that---just look in your textbook or course notes.

RGV