Question

San Francisco’s famous Steph’s Restaurant is open 24 hours per day. Servers report for duty at 3 a.m., 7 a.m., 11 a.m., 3 p.m., 7 p.m., or 11 p.m., and each works an 8-hour shift. The following table shows the minimum number of workers needed during the 6 periods into which the day is divided:

PERIOD	TIME	NUMBER OF SERVERS REQUIRED
1	3 a.m. - 7 a.m.	4
2	7 a.m. - 11 a.m.	16
3	11 am - 3 pm	19
4	3 pm - 7 pm	13
5	7 pm - 11 pm	15
6	11 pm - 3 am	7

Owner Steph Curry’s scheduling problem is to determine how many servers should report for work at the start of each time period in order to minimize the total staff required for one day’s operation. Formulate (but do not solve) the appropriate LP.

Answer 1

Let us assume the number of servers who report for duty at 3 am = X1

The number of servers who report for duty at 7 am = X2

The number of servers who report for duty at 11 am = X3

The number of servers who report for duty at 3 pm = X4

The number of servers who report for duty at 7 pm = X5

The number of servers who report for duty at 11 pm = X6

The objective function is to minimize the number of servers :

Minimum Z = X1 + X2 + X3 + X4 + X5 + X6

The shift that starts at 7 am requires a total of 16 numbers of servers. Servers who come at 7 am as well as servers who come at 3 am shall be available in this shift, and the sum total of these servers should be minimum 16.

So, X1 + X2 >= 16

Similarly,

X2 + X3 >= 19

X3 + X4 >=13

X4 + X5 >= 15

X5 + X6 >= 7

X6 + X1 >= 4

Therefore the Linear programming formulation is as follows:

Minimum Z = X1 + X2 + X3 + X4 + X5 + X6

Subject to:

X1 + X2 >= 16

X2 + X3 >= 19

X3 + X4 >=13

X4 + X5 >= 15

X5 + X6 >= 7

X6 + X1 >= 4

X1, X2 , X3, X4, X5 , X6 >=0