Dividing A Time Span Into Shifts - Over Lapping Days

I am developing a report that tracks the efficiency of industrial sorters on
an automated assembly line. The line runs around the clock in three shifts
(1: 7:30am to 3:29pm, 2: 3:30pm to 11:29pm, and 3: 11:30pm to 7:29am), and
collects certain performance indicators, such as units processed. I am
trying to calculate the units processed during each shift on each day. For
example, if a run ran from 2:30 pm to 4:30 pm, and processed 100 units, I
would want 50 units credited to shift 2 and 50 units to shift 3 (we are
assuming a steady rate of processing).

My data arrives from the machine with the following columns (each run is a
separate row):

B|D|E|G|H
Run #| Start Date| Start Time| Units Processed | End date | End Time

I have successfully used the start time to determine the starting shift with
IF statements:

=IF(--E3<0.3125,3,(IF(--E3<0.6458,1,(IF(--E3<0.9792,2,3))))) and a similar
formula to calculate the end shift. This works fine if the run only spans
two shifts within the same day.

My problem is that some of the runs, run across more than one shift and even
across days. For example, a run that ran from 10:30 pm on one day to 4:30 pm
on the next, would overlap 5 shifts and two days. I need to be able to
figure out the proportion of time spent on each shift, and use that to get
the proportion of units processed during each shift on each day.

I am currently thinking of a convoluted series of nested IF statements (and
generating lots of columns to the right of my data for each potential shift
within a run), but this seems inelegant and cumbersome. Is there a more
logical way to approach this? Even more specifically, is there a function
that can pair my start/end dates and times with a set of shift start/end
times and calculate the elapsed time for each shift within a run? Something
with MATCH or VLOOKUP maybe?