What he said, except understand there are several other perfectly qualified "Nano-PLC" options out there beside the LOGO! (the ! is part of the product name). Other people refer to them as "Smart Relays" as well. They are small stand-alone PLCs, often with a built-in programming terminal, that support maybe 6-8 inputs and 4 relay outputs in a 2" square format, some with expansion capabilities. They have numerous timimg function, counting functions, time clocks, sequencers etc. etc., more than enough to accomplish this task. The only tricky part is that they are not all available at 12VDC. I know for a fact that the LOGO! does, as well as the Allen Bradley Micro 810 (which has free PC programming software, unlike Siemens), but those are the only two I have used at 12VDC and I know that some of the cheapest ones offerred on-line are not available at that voltage. Also, be forewarned to check the temperature specs, you tend to get what you pay for.
Still, these little devices have become so inexpensive that it's a wonder to me that complex programmable timer relays even manage to continue to be made.
"You measure the size of the accomplishment by the obstacles you had to overcome to reach your goals" -- Booker T. Washington