be the number of people working on day i.
Note that such a variable definition does not match up with what we
need to find. It does us no good to know that 15 people work Monday,
13 people Tuesday, and so on because it does not tell us how many
workers are needed. Some workers will work both Monday and Tuesday,
some only one day, some neither of those days. Instead, let the days
be numbers 1 through 7 and let be the number of workers who
begin their five day shift on day i. Our objective is clearly:
Consider the constraint for Monday's staffing level of 14. Who works
on Mondays? Clearly those who start their shift on Monday ( 
).
Those who start on Tuesday ( 
) do not work on Monday, nor do those
who start on Wednesday ( 
). Those who start on Thursday ( 
)
do work on Monday, as do those who start on Friday, Saturday, and
Sunday. This gives the constraint:
Similar arguments give a total formulation of:
