Objective function for employee assignation problem solved with simulated annealing

Question

Imagine you have a total number of employees assigned to a task. Each task requires N quantity of employees for it to be 100% efficient, something like this:

Each employee is assigned to a task in this way:

Where in the metaheuristic, you make changes on the assigned tasks an employee has. Once you group by task you get the total quantity of employees currently working on the task. My question is, if you want to allocate the maximum possible quantity of workers in each task in the best possible way, which objective function will you use? I was thinking on minimizing the sum of the differences but I ran into an issue: some differences might be positive and others negative, the total sum of differences can lead to a total of zero:

Are there any other kinds of objective functions to use in this kind of problem?

Already tried the following:

• When the difference is positive, multiply the difference by a big number
• Use the absolute value of the differences
• Use a weighted sum of percentages (instead of differences)

Answer 1

I think that there is not enough information to create a optimization problem in this case. If you have more (or the same number of) employees than the needed to execute the task, all the employees have the same skills and works at the same speed, so it does not matter how you assign the employees, since you assign enough employees for each task, the solutions will be equivalent.

However, if you define the amount of work needed to finish the tasks in some standard and employee independent unit (for example man-hours or packs finisheds), define a due time to each task or different speeds to each employee, then you put one employee in one task or another can change the total number of hours, maybe cause delays or make some of them work more or less than what is defined in the contract (all those things can be individually an objective function).