The bank is working to develop an efficient work schedule for full-time and part-time tellers. The schedule must provide for efficient operation of the bank including adequate customer service, employee breaks, and so on. On Fridays the bank is open from 9:00 A.M. to 7:00 P.M. The number of tellers necessary to provide adequate customer service during each hour of operation is summarized here.

Time Number of Tellers Time Number of Tellers
9:00 a.m.-10:00 a.m. 6 2:00 p.m.-3:00 p.m. 6
10:00 a.m.-11:00 a.m. 4 3:00 p.m.-4:00 p.m. 4
11:00 a.m.-Noon 8 4:00 p.m.-5:00 p.m. 7
Noon-1:00 p.m. 10 5:00 p.m.-6:00 p.m. 6
1:00 p.m.-2:00 p.m. 9 6:00 p.m.-7:00 p.m. 6

Each full-time employee starts on the hour and works a 4-hour shift, followed by a 1-hour break and then a 3-hour shift. Part-time employees work one 4-hour shift beginning on the hour. Considering salary and fringe benefits, full-time employees cost the bank $15 per hour ($105 a day), and part-time employees cost the bank $8 per hour ($32 per day).

Required:
Formulate an integer programming model that can be used to develop a schedule that will satisfy customer service needs at a minimum employee cost. (Hint: Let xi = number of full-time employees coming on duty at the beginning of hour i and yi = number of part-time employees coming on duty at the beginning of hour i.)

It is a practical, complex scheduling problem that can be easily modelled and solved as IP. Note that part time and full time employees are paid at different rates. So, it is important that the bank operates to provide required service level (indicated by the number of tellers per hour) while minimizing total cost due to staff salary.

We use x as full time and y as part time staff.

It would help if you construct a table as shown below to better analyze the problem before attempting to model it. Red shows work hrs of full time staff and green is for part timers. Just follow their work hour rules.

So, full timers have 3 options:

(i) start at 9 – work till 1pm, take 1 hr break, then work 2 – 5 pm

(ii) start at 10 – work till 2pm, take 1 hr break, then work 3 – 6 pm

(iii) start at 11 – work till 3pm, take 1 hr break, then work 4 – 7 pm

that’s all… bank closes at 7 pm.

However, part timers don’t have 1 hr break. They work straight 4 hrs and leave. See green cells above to understand their alternatives.

Now, decision variables should include this work hour information. Here goes their definition. You must clearly define decision variables in exam, otherwise, the model is meaningless.

X9 = number of full time staff who start at 9 and follow the hour-related rule (no need to specify more details; those are already undestood)

X10 = number of full time staff who start at 10 and follow the hour-related rule

X11 = number of full time staff who start at 11 and follow the hour-related rule

Y9 = number of part time staff who start at 9 and follow the hour-related rule

Y10 = number of part time staff who start at 10 and follow the hour-related rule

…. This way you define up to Y15.

You must also specify that:

Xi, Yi >=0, integers.

Objective Function:

Minimize Z = 105*(x9 + x10 + x11) + 32* (y9 + y10 + … + y 15)

You will need one “covering” constraint for each working hour. Both full and part-timers can “cover” the duty of each hour. Look at the color-coded table (above) column-wise. The 9:00-10:00 hour can be covered only by x9 and y9 staff; next hour by x9, x10, y9, y10 only,… and so on. So, the

constraints are: (attached with image)