Question

You have the task of scheduling employees. You’ve just been handed the following information and asked to determine the number of employees that should be scheduled for their eight hour shift starting at one of the following six start times (9am, 1 pm, 5pm, 9pm, 1 am, and 5am). You’ve also been told that there needs to be a minimum number of employees during each of the following four hour time periods.

9am – 1 pm                         6

1 pm – 5pm                        7

5pm – 9pm                         9

9pm – 1 am                         8

1 am – 5am                         5

5am – 9am                          6

If an employee is scheduled at 9am, then their shift would end 8 hours later at 5pm.

Write a linear programming model (Define the variables, write the constraints, write the objective function). Do not solve.

Answer 1

Already Solved!

Let,

x1 = number of employees that start shift from 9 am-1 pm and finishes at 1pm - 5 pm
x2 = number of employees that start shift from 1pm - 5 pm and finishes at 5pm - 9 pm
x3 = number of employees that start shift from 5pm - 9 pm and finishes at 9pm - 1 am
x4 = number of employees that start shift from 9pm - 1 am and finishes at 1 am - 5 am
x5 = number of employees that start shift from 1 am - 5 am and finishes at 5 am-9 am
x6 = number of employees that start shift from 5 am - 9 am and finishes at 9 am-1 pm

objective is to minimize number of employees => Min x1+x2+x3+x4+x5+x6

subject to,

x6+x1 >= 6 (Min number of employees at 9am – 1 pm shift)
x1+x2 >= 7 (Min number of employees at 1pm – 5 pm shift)
x2+x3 >= 9 (Min number of employees at 5pm – 9 pm shift)
x3+x4 >= 8 (Min number of employees at 9pm – 1 am shift)
x4+x5 >= 5 (Min number of employees at 1am – 5 am shift)
x5+x6 >= 6 (Min number of employees at 5am – 9 am shift)

x1,x2,x3,x4,x5,x6 >= 0 (non-negativity constraint)