# What applications are there for this class of optimization problems?

 0 Consider the following optimization problem: \begin{align} \max ~~ & \sum_{i=1}^{n} f(x_i) \\ \text{subject to~~} & \sum_{i=1}^{n} x_{i} = a \\ & x_{i} \in \{0,\dots,c_i\} \forall i \in \{1,\dots,n\}. \end{align} Here, $$f(x)$$ is a non-linear function of $$x$$ and is defined for each non-negative integer $$x$$. I am interested to know where this class of optimization problems can arise in practice. Sorry, but I tried my best to use MathJax. It seems that this tutorial does not work here. asked 10 Oct '17, 08:38 Opt1989 107●7 accept rate: 0% Paul Rubin ♦♦ 14.6k●4●13 1 MathJax on this site is funky, to put it charitably. For future reference, you might find this question helpful: https://www.or-exchange.org/questions/9930/what-format-could-i-post-equations-in-or-exchange. (11 Oct '17, 15:38) Paul Rubin ♦♦

 1 The case of $$c_i=1 \forall i \in \{1, \ldots, n \}$$ resembles the best subset selection problem in regression, for a suitable loss-measuring objective function (so your maximization would need to be converted to minimization). answered 13 Oct '17, 08:55 AndyT 678●8 accept rate: 7% Paul Rubin ♦♦ 14.6k●4●13
 1 If we change the objective function to $$\min \, \sum_{i} f_i(x_i)$$, there are several potential applications in supply chain management. One is splitting an order for $$a$$ units of a part across multiple suppliers with different price schedules (which could include quantity discounts). Another is splitting a production lot of $$a$$ identical items across multiple machines (minimizing the total cost, with possible economies or diseconomies of scale on each machine). A third is splitting a load of $$a$$ items across multiple carriers (whose costs depend on load quantity). Some of those examples might make sense if all cost functions were identical (drop the subscript on $$f$$), some maybe not so much. For maximizing, maybe something like allocating $$a$$ sales people or repair people to different regions, where the objective maximizes sales made, downed power lines restored, or whatever. answered 13 Oct '17, 13:42 Paul Rubin ♦♦ 14.6k●4●13 accept rate: 19%
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