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April 24th, 2014, 07:36 PM  #1 
Newbie Joined: Apr 2014 From: US Posts: 2 Thanks: 0  Linear optimization question
I have a problem at work I'm trying to solve and I can't figure out a good way to do it, hoping someone might be able to help. I have put the relevant info in the below pastebin. Basically I want to distribute some amount of S into two bins, one of which is split into smaller bins, in such a way that the amount between the two is as close to some give ratio as it can be, and then the amount in the first larger bin is split as close as equally among the smaller bins inside of it. I was thinking about the Simplex algorithm but not sure how well it would work... Difficult Math Problem  Pastebin.com 
April 24th, 2014, 10:18 PM  #2 
Newbie Joined: Apr 2014 From: US Posts: 2 Thanks: 0 
I realize this may have been a bit vague. I'll give an example: Suppose S is the total budget say $100, S should be split into two sub groups, S_a and S_b. Sa contains sub groups, G_1 through G_n, and G_i = c_i*I_i. Think of c_i as some fixed cost per item for a group of items I_i. The second top level group, S_b, is the spillover group. Give some percentage, say r=30%, I'd ideally like 30% of the items in group S_a and 70% in group S_b. This doesn't necessarily mean the budget is split 30/70, just the items. Only so many items can fit in each subgroup( I_i <= k_i ). If not all the items can fit in S_a , put them in the S_b along with the other 70%. Basically I'm trying to figure out how many items can go in each group, keeping the ratio of items between S_a and S_b, keeping an even split of items between all subgroups G_i, and having the sum of all groups equal the total budget. 

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