1
$\begingroup$

The Carter Nut Company supplies a variety of nut mixtures for sale to companies that package and resell them. Its most expensive product is Bridge Mix, a mixture of peanuts, almonds, and cashews. Bridge Mix contains no more than 25% of peanuts and no less than 40% cashews. There is no limitation on the percentage of almonds. The current selling price of Bridge Mix is $0.80 per pound. CNC estimates that it can process up to 1000 pounds of Bridge Mix this month. The table gives the amounts of inventory and cost of the nuts to be used.

Type          Cost($per lb)         Pounds available
Peanuts           0.20                    400
Almonds           0.35                    250
Cashews           0.50                    200

So currently I am working on just setting up this Linear Programming Problem. Thus far, I have not had a problem setting these problems up, but I am hung up on this example in terms of the constraints.

My decision variables are:

x1 = number of pounds of peanuts

x2 = number of pounds of almonds

x3 = number of pounds of cashews

My objective function is: Maximize Profit

P = 0.20x1 + 0.35x2 + 0.50x3

The Constraints I understand are: (at least I think I'm right so far)

x1 + x2 + x3 ≤ 1000

And x1,x2,x3 ≥ 0 (non-negativity)

I know I am missing some constraints from the information in this question, but I am confused on how to put them together. It seems in this example there needs to be some work done to just figure out the constraints. The previous examples I have done have not required this. If I could get a couple hints here I would greatly appreciate it. Thanks.

  • 0
    You haven't addressed the constraint that no more than 25% of the mix can be peanuts. How could you formulate that in terms of your decision variables? – 2017-02-15
  • 0
    Yeah I have not addressed that one or the no less that 40% of cashews. Originally I thought something like x1 ≤ (0.25)(1000) but I'm not sure about this – 2017-02-15
  • 0
    Also how does the pounds available column play into this? – 2017-02-15
  • 0
    You obviously need to put upper bounds on the pounds of peanuts, cashews, etc. that are included in the mix since you can't buy unlimited amounts of them. – 2017-02-15
  • 0
    So this would simply be x1 ≤ 400 and so on for the rest? Was I thinking correctly about the 25% of peanuts? Thanks by the way. – 2017-02-15
  • 0
    If I read the problem correctly, your $P$ is not profit, but cost. – 2017-02-16
  • 0
    So should I say minimize cost then? – 2017-02-16
  • 1
    You should maximize profit, which is equivalent to minimizing cost if you know you are going to sell 1000 pounds of the Bridge Mix. But it's better to take the safe road and compute profit as revenue minus cost. The revenue is $(x_1+x_2+x_3)\cdot 0.8$. – 2017-02-16
  • 0
    Ok that makes sense. How about the constraints? It says no less than 40% cashews but they only have 200 pounds available. There is no way they can make 1000 pounds if at least 40% are cashews based on what they have. I don't understand how they can think they can make 1000 pounds – 2017-02-16
  • 1
    Well, they will make no more pounds than the constraints allow them to make. You should expect the solution to have exactly 40% cashews. – 2017-02-16

0 Answers 0