Formation and solving procedure (Optimization)

Question

There are $n_1, n_2, n_3, ..., n_n $companies are there.

Say total number of companies in a region is $'n'$ , where $'n'$ is some positive integer.

The average no. of employees working in each company is $'k'. $

For example,

the no. of employees working in first company $n_1$ is $4$ (say) and in $n_2$ is $3$ (say), ..., $n_n$ is $48$ (say)

$Note:$

1) $4, 3, ..., 48$ are all obviously positive integers. Thus, the average no. of employees in all companies are $'k'$ (say), also positive integer.

2) The least number of employees in any company is $3$ and maximum no of employees is $2000$

$Restriction:$ Every employee has to work 7 days in other company or companies in year is mandatory.

For example, $k_1$ is one of the employee in company $n_{13}$

Then, $k_1$ has to work $7$ days in others company (other than his parent company $n_{13}$).

$Note:$

This $k_1$ can work $2 $days in company $n_1$; $1$ day in $n_{12}$ and $3$ days in $n_{100}$ and $1$ day in $n_{23}$.

This means, $k_1$ can work in single slot of consecutive of $7$ days in any company other than his parent company or he can work in different days ( all together $7$ days) in a year.

Now the question is: "How to formulate the entire problem, by keeping the following in mind"

1. All employees has to work in other companies either consecutive of $7$ days or different days in different companies all together $7$ days in year other than his/her parent company.

2. One or more employee of one company can opt work in other companies in single or different slots are allowed. For example: $n_2$ has $3$ employees. all three or any one can opt work in other than company $n_2.$

3. least no of employees are $3$ and maximum employees are $2000.$
4. All companies and all employees are in consideration of the cited above restrictions.

With respect to the above information, without leaving one employee and without leaving one company, how best we can formulate? and how many combinations we can create?

I hope the edited post will give some clarity. Kindly help in formulating the problem and solving in general.

Thank you all.