I have the following problem: Consider the following scheduling problem for a factory that operates 24 hours per day, 7 days per week. In a given day, there are requirements for the total number of employees that must be at the plant. These are given in the following table:
Hours Employees needed
12 a.m. to 4 a.m. 8
4 a.m. to 8 a.m. 10
8 a.m. to 12 p.m. 16
12 p.m. to 4 p.m. 21
4 p.m. to 8 p.m. 18
8 p.m. to 12 a.m. 12
Employees can work either 8-hour or 12-hour shifts, starting at the times stated above; the 12-hour shifts can only start at 12 a.m/p.m. or 8 a.m./p.m. Those working the 8-hour shifts cost the company $40 per hour, and those working 12-hour shifts cost the company $60 per hour. How should the company staff the plant so as to minimize labor costs?
I have come up with the following:
xi = number of employees working an 8 hour shift starting at time i
xj = number of employees working a 12 hour shift starting at time j
t1 = 8am-12am shift
t2 = 12am-4am shift
t3 = 4am-8am shift
t4 = 8am-12pm shift
t5 = 12pm-4pm shift
t6 = 4pm=8pm shift
t7 = 8pm-12am shift
then I have the following for 8 hour shifts
t1+t2 <= 8+10
t2+t3 <= 10+16
t3+t4 <= 16+21
t4+t5 <= 21+18
t5+t6 <= 18+12
t6+t1 <= 12+8
for the 12 hour shifts I have:
t1+t2+t3 <= 8+10+16
t3+t4+t5 <= 16+21+18
t4+t5+t6 <= 21+18+12
t6+t1+t2 <= 12+8+10
I also have the following constraint:
t1,t2,t3,t4,t5,t6>= 0 and integers
Then I have come up with the following minimizing function:
40(t1+t2) + 40(t2+t3) + 40(t3+t4) + 40(t4+t5) + 40(t5+t6) + 40(t6+t1) + >60(t1+t2+t3) + 60(t3+t4+t5) + 60(t4+t5+t6) + 60(t6+t1+t2)
Can anyone verify if I have modeled this correctly? Thank you!!