algorithm - A new Bin-packing? -


i'm looking in kind-of bin-packing problem, not quite same. problem asks put n items minimum number of bins without total weight exceeding capacity of bins. (classical definition)

the difference is: each item has weight , bound, , capacity of bin dynamically determined minimum bound of items in bin.

e.g., have 4 items a[11,12], b[1,10], c[3,4], d[20,22] ([weight,bound]). now, if put item bin, call b1, capacity of b1 become 12. try put item b b1, failed because total weight 11+1 =12, , capacity of b1 become 10, smaller total weight. so, b put bin b2, capacity become 10. now, put item c b2, because total weight 1+3 =4, , capacity of b2 become 4.

i don't know whether question has been solved in areas name. or variant of bin-packing has been discussed somewhere. don't know whether right place post question, helps appreciated!

usually algorithm design np-hard problems, it's necessary reuse techniques rather whole algorithms. here, algorithms standard bin packing use branch-and-bound column generation carry on well.

the idea formulate enormous set cover instance sets sets of items fit single bin. integer programming technique normal set cover, there many sets need else, i.e., column generation. there one-to-one correspondence between sets , columns, rip out part of linear programming solver uses brute force find column enter , replace solver turns out knapsack analog of problem.

this modified knapsack problem is, given items weights, profits, , bounds, find profitable set of items total weight less minimum bound. dynamic program solving knapsack small integer weights happily transfers on no loss of efficiency. sort items descending bounds; then, when forming sets involving recent item, weight limit item's bound.


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