Larry Edison is the director of the Computer Center for Buckly college. He now needs to schedule the staffing of the center. It is open 8 a.m. until midnight, Larry has monitored the usage of the center at various times of the day, and determined that the following number of computer consultants are required:

Two types of computer consultants can be hired: full-time and part-time. The full-time consultants work for 8 consecutive hours in any of the following shifts: morning (8 A.M.–4 P.M.), afternoon (noon–8 P.M.), and evening (4 P.M.–midnight). Full-time consultants are paid $40 per hour. Part-time consultants can be hired to work any of the four shifts listed in the above table. Part-time consultants are paid $30 per hour. An additional requirement is that during every time period, there must be at least 2 full-time consultants on duty for every part-time consultant on duty. Larry would like to determine how many full-time and how many part-time workers should work each shift to meet the above requirements at the minimum possible cost.

Required:
Formulate a linear programming model for this problem.

From the question, we are told that full time consultants need $40 per hour while part time consultants would need $30 per hour. Since part time require lesser, then it is smart to hire more part time consultants than full time.

But we are told that at every work time period, there must be at least 2 full time consultants on duty for every part time consultant that's on on duty.

Since full time consultants work 8 consecutive hours, the company will need a minimum of 4 full time consultant which is 2 in the morning from 8am till 4pm and another 2 in the evening from 4pm till midnight.

Now, the cost for full time consultant is;

C_full = no of full time consultant × hours each consultant work × hourly rate

Number of full time consultants in a day = 4

Hours each works = 8 hours

Hourly rate = $40

Thus;

C_full = 4 × 8 × 40

C_full = $1280

Since we are taking part time consultant for the remaining number of consultants, then;

for time period of 8 am - noon from the table, we need 4 minimum number of consultants. This means, we need to hire; 2 full time consultants and 4 - 2 = 2 part time consultants

for time period of noon till 4pm from the table, we require 8 minimum number of consultant. Thus, we need so we need to hire 2 full time consultants and 8 - 2 = 6 part time consultants.

for time period of 4 pm till 8 pm from the attached table, we require 10 minimum number of consultants. Thus, we need to hire 2 full time consultants and 10 - 2 = 8 part time consultant

for time period of 8 am till midnight from the table attached, we require 6 minimum number of consultant. Thus, we need to hire 2 full time consultants and 6 - 2 = 4 part time consultants.

Now, let's Calculate total cost for part time consultants;

C_part = no of part time consultants × hours each consultant works × hourly rate

C_part = (4 + 8 + 6 + 2) × 4 × 30

C_part = 20 × 4 × 30

C_part = $2400

Thus;

minimum cost =C_full + C_part

Minimum cost = $1280 + $2400

Minimum cost = $3680