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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.

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  • $\begingroup$ Your cost/availability has three lines of almonds. Clearly two should be other nuts. $\endgroup$ – Ross Millikan Mar 11 '16 at 4:44
  • $\begingroup$ I'm not sure I understand where the three lines of almonds you mentioned is coming from $\endgroup$ – Max Mattappillil Mar 11 '16 at 5:23
  • $\begingroup$ In your info on the price to buy each nut and how much is available. $\endgroup$ – Ross Millikan Mar 11 '16 at 5:27
  • $\begingroup$ Jeez my brain is fried to the point where i'm not even noticing simple errors like that. I don't think there are any more errors like that in the question $\endgroup$ – Max Mattappillil Mar 11 '16 at 5:33
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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.

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  • $\begingroup$ It's my fault for not mentioning the other assortments. I wanted to get a better understanding of how to approach the problem for the Regular assortment so that I could apply the same understanding towards the other assortments. I also did not want to put the other assortment requirements here because I did not want to make it seem like I was just asking for an answer. I don't think knowing what the other assortment requirements are will be needed for finding the maximum profit for the regular assortment however $\endgroup$ – Max Mattappillil Mar 11 '16 at 4:14
  • $\begingroup$ I'll add in the other requirements to complete all of the information for the problem $\endgroup$ – Max Mattappillil Mar 11 '16 at 4:17
  • $\begingroup$ I think you missed the point of the problem if you thought the first post could help. You have to decide how many pounds of each mix to make (at an acceptable mix). Since the Blue Ribbon sells at much more per pound than the others, the naive thing is to make as much of that as you can, then see what is left to make the others. This can fail to be optimum if reducing the quantity frees up scarce resources (cashews and almonds) so you can make lots of one of the others. $\endgroup$ – Ross Millikan Mar 11 '16 at 4:43
  • $\begingroup$ Your profit is revenue minus cost. The revenue is the 1.80/lb times the amount of Blue Ribbon made plus the other two. The cost is the total cost of all the nuts you use. You want to optimize profit. Your $X$s should be the weight of each nut in each mix. You have constraint equations coming from the quality specs and the total amount of each nut available. The cost constraints are correct but not useful-it would be better to do them in pounds. $\endgroup$ – Ross Millikan Mar 11 '16 at 5:48

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