Integer programming problem.

Question

I do not know the integer programming problem. I am reading a paper (1 below) which makes the following claim for the problem P1. I have two questions regarding this claim which I will post after posting the problem from the paper.

Problem (P1) is a three-dimensional integer programing problem whose solution space is in the size of $2^{NM(K+2)}$.

My questions are: 1. Why is the problem P1 a 3D problem? P1 is given in the image with this post. 2. Why is the solution space $2^{NM(K+2)}$? I understand that without know N,M, or K no one can give me an answer about this solution space. But I want to know how is the size of the solution space found by the authors?

$[1]$ Multi-Server Multi-User Multi-Task Computation Offloading for Mobile Edge Computing Networks

Answer 1

1. It's a 3D problem because $a_{nmk}$ has three subscripts, so that $a\in \lbrace 0,1 \rbrace^{N\times M\times K}$ where $N=|\mathcal{N}|$ etc.
2. Assuming that everything other than the $a$ variables is a constant, I don't know where they get $2^{NM(K+2)}$ for the size of the solution space. Ignoring (13c), the size of the solution space would be $2^{NMK}$. Assuming that (13c) holds for all $m\in\mathcal{M}$ and $n\in\mathcal{N}$ (the presentation is mathematically sloppy), the feasible portion of the solution space would have size $K^{NM}$. (For each combination of $n$ and $m$, you would have $K$ ways to satisfy (13c), and choices for one $n,m$ pair would be independent of choices for other pairs.)