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I've got problem on integer programming, specifically with the following knapsack problem. I'd be happy to get some suggestions on how to solve the problem in a time efficient way.

There are 120 items $i$, each with weight $w_i$ and the value $v_i$. Each item can be selected only once (or zero times). For each of them one has to chose a quality level ($a$, $b$, $c$, $d$ or $e$).
The higher the level, the higher is its weight and its value. Both are precalculated for all items and stored in matrix.

Now the question is, given a capacity $W$, how to get the optimal levels for each of the items. Since the item set is relatively huge, the aspects of computational complexity might be of special relevant, I guess.

Can you tell me which algorithms can be use for this variant of the knapsack problem? Thanks a lot!

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    $\begingroup$ welcome to MO! I find your question interesting, but I would like to see some clarification regarding the quality levels: what do they affect? You write "the higher the level, the higher its weight and value"; are weight and value the same or different properties that happen to have equal value? If you could provide a linear programming formulation of your problem, that would surely enhance your question and more likely generate the feedback you are looking for. $\endgroup$ – Manfred Weis Oct 7 '18 at 12:49
  • $\begingroup$ @ManfredWeis I conjecture that the "weights" are the things whose total is to be bounded by the "capacity" $W$ and that the "values" are the things whose total is to be maximized for "optimum levels". I agree with you that this (or whatever connection the OP intended between weights, values, capacity, and optimization) should be specified in the question. $\endgroup$ – Andreas Blass Oct 7 '18 at 19:12
  • $\begingroup$ Have you considered moving this question to cs.stackeschange.com? $\endgroup$ – S. Carnahan Oct 8 '18 at 16:40
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If you set both the values and the weights of each item corresponding to the lowest four levels to $0$, this problem is exactly the standard knapsack problem, so it is no easier than the standard knapsack problem, i.e. it is NP-hard.

You can use the dynamic programming algorithm for the standard knapsack problem to solve this problem, except that you should consider six cases instead of two: the item is not chosen, or its level $a,b,c,d$ or $e$ is chosen, then take the maximum value. The complexity is $O(nmW)$ where $m$ is the number of levels.

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Your basically want to solve a variant of the 0-1 Knapsack problem. Your variant includes the following changes to the standard problem: for each item instead of one set of $[wi, vi]$, you have five sets corresponding to each of the quality levels.

Normal 0-1 Knapsack can be solved using dynamic programming in $O(nW)$ time and $O(nW)$ space where $n$ is the number of items and $W$ is the size of the Knapsack.

In your case, the complexity will be $O(nqW)$ time and $O(nqW)$ space where $q$ is the number of quality levels. If $q << n$, then we are back to $O(nW)$ time and $O(nW)$ space.

Refer this handout for a quick brush-up on the actual recurrence relation.

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