I have two type of resources A and B. The resources are to be distributed (discretely) over k nodes.
the number of resources A is a
the number of resources B is b
resources B should be completely distributed (sum of resources taken by nodes should be b)
resources A may not be completely distributed over the nodes. In fact, we want to reduce the usage of resources A.
Given resources (A or B) to a node enhance the quality of the nodes, where the relation is non-linear.
All nodes should achieve a minimum quality.
What is the type of the problem and how I can find the optimal value?
Can also use Grasshopper Optimization Algorithm with constraint to be minimized or maximized.
Bet the problem statement requires optimal solution with the goal to minimize or maximize constraint....Genetic Algorithm as stated by @Alessandro Ferrarini can handle this case. If you are adept with softwares tools for this options.
Genetic algorithms never find an optimum. I hope you will use better tools than those. You should know that this forum has basically been hi-jacked by meta-heuristics enthusiasts - an inflated array of posts at RG are based on the fact that RG "scholars" do not know that there are better tools. So beware!
You describe a MINLP with non-linear constraints (the minimum quality for each node) and a linear objective (if the only criterion is to minimize the usage of A).
Depending on the properties of the constraint functions (convexity of the feasible region?) a MINLP solver like the Zuse Institute optimization suite using Ipopt will find the global optimum or a global solver like BARON could be needed.
My choice would be to use GAMS for modeling and test different solvers for that type of model available with GAMS. Your problem seems to give a rather small problem (if the number of nodes is not too big), then the free trial version of GAMS would be sufficient.
It seems to be a mixed-integer (MINLP) bin-packing problem. Such problems end up being NP hard.
"Given resources (A or B) to a node enhance the quality of the nodes, where the relation is non-linear." Can you describe the relation? Is it possible to use McCormick envelopes to linearize it? I assume you have only binary variables. Is it true?
$$\min \sum_{j\in Nodes}\sum_{ik\in B\times A} y_{ik}.f(x_{jk}^{a}, x_{ji}^{b})$$
$$ \sum_{j\in Nodes} x_{ji}^{b} = 1, \quad\forall i\in B $$
$$ \sum_{j\in Nodes} x_{jk}^{a} \leq 1, \quad\forall k\in A $$
Some think like this.
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