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 xijx_{ij} be a binary decision variable that is 1 if task tjt_j is assigned to node nin_i and 0 otherwise.

Minimize the total cost:

Minimizei=1mj=1kxijCij{Minimize} \sum_{i=1}^{m} \sum_{j=1}^{k} x_{ij} \cdot C_{ij}

Subject to the computational demand and capacity constraints:

i=1mxijDjCapacityi, i\sum_{i=1}^{m} x_{ij} \cdot D_j \leq \text{Capacity}_i, \ \forall i
j=1kxij=1, j\sum_{j=1}^{k} x_{ij} = 1, \ \forall j

Where CijC_{ij} is the cost of assigning task tjt_j to node nin_i, and Capacityi{Capacity}_i is the computational capacity of node nin_i.