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 $n_i$.