# Allocation Optimization

The objective is to minimize the total processing time or cost while ensuring that the computational demand of tasks is met by the allocated nodes. Let $$x\_{ij}$$ be a binary decision variable that is 1 if task $$t\_j$$ is assigned to node $$n\_i$$ and 0 otherwise.

Minimize the total cost:

$$
{Minimize} \sum\_{i=1}^{m} \sum\_{j=1}^{k} x\_{ij} \cdot C\_{ij}
$$

Subject to the computational demand and capacity constraints:

$$
\sum\_{i=1}^{m} x\_{ij} \cdot D\_j \leq \text{Capacity}\_i, \ \forall i
$$

$$
\sum\_{j=1}^{k} x\_{ij} = 1, \ \forall j
$$

Where $$C\_{ij}$$ is the cost of assigning task $$t\_j$$ to node $$n\_i$$, and $${Capacity}\_i$$ is the computational capacity of node $$n\_i$$.