Dear Sir

It appears to be quite related to the generalized assignment problem, GAP (or, capacitated assignment problem, as it may be called): you have assignment constraints, and in fact two capacity constraints, whereas GAP has only one. On the other hand your objective function is simpler in that you have common constant in one of the dimensions. It still appears to me that you have a problem that is more complicated than the GAP. 

Dear Mr. Patriksson, thank you for the help. I have updated the figure with the optimization problem with some properties of the constants of the problem that may help in answering my question.

While the value of a_m grows with m, making higher indices m be better than lower ones in terms of objective value, the same is true for how much capacity is used, so there does not appear to be anything directly useful in that information. (Had it been the other way around there might have been some greedy procedure one might be able to use in order to reduce the number of active variables.) As fas as I can see, after two minutes :-), there does not seem to be a favorable structure here. But another five minutes, maybe.

How big problems do you want to solve? Do you somehow want to learn about some relationships among the variables in an optimal solution, as functions of the constants? I do not think that is possible.

Thank you again for the help. I will search for extensions of GAP problem in the literature. 

This looks like a two-dimensional multiple-choice knapsack problem.  It could be solved in pseudo-polynomial time with dynamic programming.

(I don't know whether one could exploit the special structure of the coefficients to obtain a polynomial-time algorithm.)

Thank you Adam. Do you have any reference about that?

See this old book:

http://www.or.deis.unibo.it/knapsack.html

or this more recent one:

http://www.springer.com/mathematics/book/978-3-540-40286-2

especially Chapter 9.

Thank you again, Adam.

Dear Sir