Profit Maximization vs Cost Minimization for Employee Scheduling

Question

I wanted to write two objective functions for an employee scheduling problem (MIP) until it occurred to me, that one objective function may be redundant.

• Is there a difference between the cost minimization and profit maximization for this assigning problem?

In the following, I have made an example in Excel and the values and assigning, seems to be the same.

All 4 employees are available and capable of taking all 3 projects.

Noted: I made the assigning manually.

The way I chose the employees for the cost minimization objective is the following:

• Get the project with the most days to fill within a week.
• Assign the most cost-effective employee to this project.
• Repeat this process until all projects are fulfilled.

For the profit maximization:

• Get the daily revenue from a project
• Subtract it from the employee's daily cost
• Multiply it with the days in a week the employee should work on this project.
• Assign the employees such the sum from all calculations such the above is maximum.

The optimum assigning seems to be Cost Minimization v1 and Cost Minimization v2 which would also work with the profit maximization.

Furthermore, all projects must be staffed and the achieved revenue is not employee dependant.

Answer 1

If you assume that (a) all projects must be fully staffed and (b) the revenue for a project is constant regardless of who does it, then total revenue is fixed. In that case, cost minimization and profit maximization are the same. Where they would differ would be if either not all projects need be done (but some minimum number must be -- otherwise cost minimization is achieved by doing nothing) or project revenue is a function of who does it (in which case using the cheaper labor might "leave money on the table" in excess of the cost savings).