Maximizing the value of a set of item pairs given limited numbers of items [closed]

I have three types of items: $a_1$, $a_2$ and $a_3$, each with quantities $c_1$, $c_2$, and $c_3$. I can combine one of each of two different types of items (for example combining $a_1$ with $a_3$) into a product $p_{ij}$ which has a value of $v_{ij}$. The value of each product is not related to the individual items, but specifically to the combination of those items. With 3 types of items, there are only 3 possible products, but the number of products increase when given a generalized number of item types $a_n$.

How do I maximize the total value all items paired into products, given limited and differing quantities for each item?

I have been struggling for a way of formally conceptualizing this problem in order to find an algorithm that may work. I've spent time learning about knapsack and generalized assignment problems to see if I can fit this problem into one of those to no avail. A greedy algorithm will, of course, find a local max, and I will probably end up using that if I can't figure out anything better.

closed as unclear what you're asking by Namaste, Daniel W. Farlow, José Carlos Santos, Claude Leibovici, user91500Jul 26 '17 at 9:00

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• Have you had a look at linear programming? – Jens Jul 25 '17 at 23:08
• I had not. I think that's what I'm looking for: Maximize:$$v_{12}p_{12}+v_{13}p_{13}+v_{23}p_{23}$$Subject to:$$a_{11}p_{12}+a_{12}p_{13} \le a_1$$$$a_{21}p_{12}+a_{22}p_{23}\le a_2$$$$a_{31}p_{13}+a_{32}p_{23} \le a_3$$ – Rek Jul 26 '17 at 0:54
• Glad to help! If you need additional guidance, I suggest asking a new question outlining the problem, where you include the tag "linear-programming". I'm not an expert on the topic, but others are. – Jens Jul 26 '17 at 1:14
• Will do, thanks. – Rek Jul 26 '17 at 1:27