Suppose we are trying to maximize an objective function defined as such:
$$f(\lambda, \mathcal{S}) = \sum_{i\in\mathcal{S}} v_i - n\lambda\pi\max_{(i,j)\in \mathcal{S}}(d(x_i,x_j))^2/4$$
where $\mathcal{S}$ denotes the set of chosen cities, $d(x,y)$ denotes the Euclidean distance between cities at positions $x$ and $y$, and $\lambda$ is a parameter indicating the importance attributed to the deployment cost. Likewise, we will denote by $$\mathcal{S}^\ast = \arg\max_{\mathcal{S}\in\mathcal{P}(1,\dots,n)} f(\lambda, \mathcal{S})$$
And $\{v_i\}$ and $\{x_i\}$ are the 'values' and positions of the cities distributed as such $\mathcal{U}[0,1]$ and $\mathcal{U}[0,1]\times\mathcal{U}[0,1]$ respectively.
Is it possible to analytically find (or approximate) $\mathbb{E}(|\mathcal{S}|)$ and more importantly $\mathbb{E}(f(\lambda, \mathcal{S}^\ast))$.
So I thought we could maybe model the number of these cities in a certain region of our 2D plan of size (1x1) be modeled as a spatial point (Poisson) process? and if we can do that, could we find the expected maximum number of cities in a 2D square window/ region of size (b x b)?
If all of that is possible we could then just maximize this expectation minus the cost (the expression with minus lambda) with respect to b and get an approximate answer to the problem, right?
