The problem is the following following:

```
N = {1,...,m} - servers;
Q = {q{1},...,q{m}} - the required amount of resources for tasks, where qi is the required amount of resources for task i;
T = {1, ..., k} - tasks;
V = {v{1}, ..., v{k}} - the amount of resources on servers, where vi is the amount of resources on server i;
DA - the matrix with the amount of data transferred between tasks. DA{i,j} is the amount of data transferred between task i and task j;
NC - the matrix with the network speeds between servers. NC{i,j} is the network speed between server i and server j;
IC - the matrix with a penalty for assigning two tasks on the same server. IC{i,j} is the penalty for assigning task i and task j on the same server;
x{i,j} = 0 if task i is assigned on server j and x{i, j} = 1 if task i is not assigned on server j
```

I want to minimize the total communication time and minimize the total penalty in two steps. The first step is to minimize the total penalty and the second step is to minimize the total communication time with the penalty constraint.

Total communication time:

Total penalty:

Constraints:

each task is assigned to exactly on server:

the total required amount of resources for tasks on servers does not exceed the available amount of resources on the servers:

and:

Hence the optimization problem is:

To make the objective function linear, I add constraints z{t1,t2,n1,n2} >= x{t1,n1}+x{t2,n2}-1, z{t1,t2,n1,n2} <= x{t1,n1}, z{t1,t2,n1,n2} <= x{t2,n2}, 0 <= z{t1,t2,n1,n2} <= 1

First step:

Let min_penalty be the result of the first step, then the second step:

Is everything correct?

P.S. Sorry for the grammar mistakes and formatting.