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POHow can I efficiently find the subset of activities that stay within a budget and maximizes utility?
 

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  1. POHow can I efficiently find the subset of activities that stay within a budget and maximizes utility?
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    <p>I am trying to develop an algorithm to select a subset of activities from a larger list. If selected, each activity uses some amount of a fixed resource (i.e. the sum over the selected activities must stay under a total budget). There could be multiple feasible subsets, and the means of choosing from them will be based on calculating the opportunity cost of the activities not selected.</p> <hr> <p><strong>EDIT:</strong> There are two reasons this is not the <a href="http://en.wikipedia.org/wiki/Knapsack_problem#0-1_knapsack_problem" rel="nofollow">0-1 knapsack problem</a>:</p> <ul> <li>Knapsack requires integer values for the weights (i.e. resources consumed) whereas my resource consumption (i.e. mass in the knapsack parlance) is a continuous variable. (Obviously it's possible to pick some level of precision and quantize the required resources, but my bin size would have to be very small and Knapsack is <code>O(2^n)</code> in W.</li> <li>I cannot calculate the opportunity cost a priori; that is, I can't evaluate the fitness of each one independently, although I can evaluate the utility of a given set of selected activities or the marginal utility from adding an additional task to an existing list.</li> </ul> <hr> <p>The research I've done suggests a naive approach:</p> <blockquote> <p>Define the <a href="http://en.wikipedia.org/wiki/Power_set" rel="nofollow">powerset</a><br> For each element of the powerset, calculate it's utility based on the items not in the set<br> Select the element with the highest utility</p> </blockquote> <p>However, I know there are ways to speed up execution time and required memory. For example:</p> <ul> <li>fully enumerating a powerset is <code>O(2^n)</code>, but I don't need to fully enumerate the list because once I've found a set of tasks that exceeds the budget I know that any set that adds more tasks is infeasible and can be rejected. That is if <code>{1,2,3,4}</code> is infeasible, so is <code>{1,2,3,4} U {n}</code>, where n is any one of the tasks remaining in the larger list.</li> <li>Since I'm just summing duty the order of tasks doesn't matter (i.e. if <code>{1,2,3}</code> is feasible, so are <code>{2,1,3}</code>, <code>{3,2,1}</code>, etc.). </li> <li>All I need in the end is the selected set, so I probably only need the best utility value found so far for comparison purposes.</li> <li>I don't need to keep the list enumerations, as long as I can be sure I've looked at all the feasible ones. (Although I think keeping the duty sum for previously computed feasible sub-sets might speed run-time.)</li> </ul> <p>I've convinced myself a good recursion algorithm will work, but I can't figure out how to define it, even in pseudo-code (which probably makes the most sense because it's going to be implemented in a couple of languages--probably Matlab for prototyping and then a compiled language later). </p>
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    1. USAdam Wuerl
    1. USAdam Wuerl
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      1. POHow can I efficiently find the subset of activities that stay within a budget and maximizes utility?
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    1. COunfortunately, this is NP-complete: http://en.wikipedia.org/wiki/Knapsack_problem so it's unlikely that you will find a fast algorithm.
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      1. POHow can I efficiently find the subset of activities that stay within a budget and maximizes utility?
      1. USHenrik
    2. CO@Henrik: You information is great, but note that the OP only want an exponential algo which is fine.
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      1. POHow can I efficiently find the subset of activities that stay within a budget and maximizes utility?
      1. USZiyao Wei
    3. CO@Adam How large is the set? 2^n is HUGE, even after optimizing it could be terrible. Also, I find your problem similar to one that I've answered before: http://stackoverflow.com/questions/6629581/which-optimization-algorithm-should-i-use-for-maximizing-profit-with-time-limitat/6629641#6629641
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