When the city investigated this same move 20 years ago, they determined that there would be NO UPFRONT SAVING at all. The contract would cost about the same as paying the employees. The entire incentive to move forward with the project was (1) to eliminate the pension costs which would save millions, and (2) reduce exposure to insurance liability costs, because the workers kept filing questionable injury claims against the City. There is always the threat that a claim would exceed the city's insurance policy, and the city would have to shell out money.
To be perfectly honest, I think the city is not going far enough. I would privatize almost the entire city government, including the Health Dept and code enforcement.