Earlier quoted context omitted.
Im in the 'wait and see' camp.. but you can say a program is voluntary, while in truth it's required to keep your job. Just label all the people who refuse as not being team players. Dock them on their performance reviews for it. "Johnny over there wouldn't take a short detour on his way home, so we had to send someone out to deliver a package. If he's not willing to help out, maybe he should go work some place else.…
Do you have any evidence at all that this will be the case here?
Once you have a sheet with times and delivery points for each employee.. it becomes a measure of the employees willingness to assist. If other employees are willing to drive 10 minutes, and you're not willing to drive 2 minutes, it raises the question of why.
And that question is definitely going to be asked if the alternative (30 minutes for regular delivery) results in an additional cost to the company.
But as I said, we'll have to wait and see.