Below is a problem on Linear Programming. I think I have a start to it, but I'm in a rut right now so I was wondering if I could receive a little help. Here's the problem
A candy store sells three different assortments of mixed nuts containing varying amounts of almonds, pecans, cashews, and walnuts. To preserve the store's reputation for quality, certain maximum and minimum percentages of the various nuts are required for each type of assortment, as shows in the following table.
Nut Assortment: Regular
Requirements: Not more than 20% cashews, not less than 40% walnuts, not more than 25% pecans, no restriction on almonds
Selling price per pound: $0.89
Nut Assortment: Deluxe
Requirements: Not more than 35% cashews, not less than 25% almonds, no restriction on walnuts and pecans
Selling price per pound: $1.10
Nut Assortment: Blue Ribbon
Requirements: Between 30% and 50% cashews, not less than 30% almonds, no restriction on walnuts and pecans
Selling price per pound: $1.80
Below is the cost per pound for each nut as well as the max quantity available per week
Almonds: $0.45/pound. 2000 lbs available per week
Pecans: $0.55/pound. 4000 lbs available per week
Cashews: $0.70/pound. 5000 lbs available per week
Walnuts: $0.50/pound. 3000 lbs available per week
The store would like to determine the exact amounts of almonds, pecans, cashews, and walnuts that should go into each weekly assortment to maximize its weekly profit.
Here's what I have so far:
X1 = The assortment of cashews in the regular mix in pounds
X2 = The assortment of walnuts in the regular mix in pounds
X3 = The assortment of pecans in the regular mix in pounds
X4 = The assortment of almonds in the regular mix in pounds
X5 = The assortment of cashews in the deluxe mix in pounds
X6 = The assortment of almonds in the deluxe mix in pounds
X7 = The assortment of walnuts in the deluxe mix in pounds
X8 = The assortment of pecans in the deluxe mix in pounds
X9 = The assortment of cashews in the blue ribbon mix in pounds
X10 = The assortment of almonds in the blue ribbon mix in pounds
X11 = The assortment of walnuts in the blue ribbon mix in pounds
X12 = The assortment of pecans in the blue ribbon mix in pounds
Cost Constraints (obtained from multiplying cost per pound by max quantity)
0.45(X4+X6+X10) ≤ $900 per week
0.55(X3+X8+X12) ≤ $2200 per week
0.7(X1+X5+X9) ≤ $3500 per week
0.5(X2+X7+X11) ≤ $1500 per week
Objective Function
This is where I need some help now after thinking about this problem. I can't figure out how to tie in the selling price per pound with the cost per pound for each specific nut. Could anyone provide some clarification? Many thanks.
Best Answer
Your problem is missing a lot of information. You say there are three mixes, but there is no information about how much they can sell of each mix based on composition. In that event you have to assume you can sell all you can make, so make just one mix that uses up all of at least one component. As you make a profit on every nut sold, sell all the walnuts and all the almonds, then all the cashews and pecans you can without violating the composition restraints.