Question

The Bank of Seattle is open Monday through Friday from 10 a.m. to 5 p.m. From past experience, the bank knows that it needs the number of tellers shown in the table below at various times of the day.

The Bank of Seattle hires two types of tellers. Full-time tellers work 10 a.m. to 5 p.m., 5 days a week, with 1 hour off each day for lunch. The bank determines when a full-time employee takes his or her lunch hour, but each teller must go between 11 a.m. and noon, or noon and 1 p.m., or between 1 p.m. and 2 p.m. Full-time employees are paid $22 per hour, including a paid lunch hour.

The bank can also hire part time tellers. Each part-time teller works exactly 3 consecutive hours each day, the same time every day of the week. A part-time teller is paid $14 per hour. To maintain adequate quality of service, the bank has decided that at most 5 part-time tellers can be hired, and that at all times there must be at least 2 full-time tellers working for each part-time teller. Formulate (but not to solve) the problem as an LP in order to meet the bank’s teller requirements at the minimum cost.

Time of Day	Minimum # of Tellers Required
10 am – 11 am	10
11 am – 12 pm	7
12 pm – 1 pm	6
1 pm – 2 pm	5
2 pm – 3 pm	8
3 pm – 4 pm	10

Answer 1

Solution:

Let X11, X12 and X1 are the number of full time teller that takes their lunch at 11 am, 12 noon and 1 PM respectively.

Let Yi be the part time tellers that starts their shift at i=10, 11, 12, 1, 2.

Since full time employees are paid 22$ per hour therefore for 7 hours (10 – 5) payment will be 22x7 = $154. Similarly, part time tellers would be paid 14x3=$42.

Let us assume that from 4PM to 5 PM, bank requires 10 numbers of tellers.

Following is the linear programming problem

Z = 154(X11 + X12 + X1) + 42 (Y10 + Y11 + Y12 + Y1 + Y2)

Subject to,

X11 + X12 + X1 + Y10 ≥ 10                                                           ------------- AT 10 AM

X12 + X1 + Y10 + Y11 ≥ 10                                                            ------------- AT 11 AM

X11 + X1 + Y10 + Y11 +Y12 ≥ 10                                                   ------------- AT 12 NOON

X11 + X12 + Y11 + Y12 + Y1 ≥ 10                                                  ------------- AT 1 PM

X11 + X12 +X1 + Y12 + Y1 + Y2 ≥ 10                                            ------------- AT 2 PM

X11 + X12 +X1 + Y1 + Y2 ≥ 10                                                      ------------- AT 3 PM

X11 + X12 +X1 + Y2 ≥ 10                                                             ------------- AT 4 PM

Where ,

Y10 + Y11 + Y12 + Y1 + Y2 ≤ 5

And

X11, X12 , X1, Y10, Y11 ,Y12, Y1 ,Y2 ≥ 0