Function to Determine Plunger Force on Solenoid Actuator
In general, when designing a system like this the easiest thing is to simply look at the Stroke/Force Curves for each solenoid published on manufacturer's datasheets. You will soon get a good feeling for the Power that you require.

One thing to bear in mind is duty cycle. You can put 48V through a 12V solenoid, (in order to boost your force at a high extension) but, because it will be drawing 16x more power than usual, you can only keep 48V through it for a very short period of time to avoid overheating. Again - the limits on this are all published on datasheets.

This means that for some applications, you may be able to use a high-voltage pulse to 'pull in' your load from a long stroke position, and then step down to the continuously-rated voltage in order to hold the load at zero stroke.

For a more theoretical approach, have a read of this: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.428.9647&amp;rep=rep1&amp;type=pdf