dolution

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 from 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:

Time of Day	Minimum Number of Consultants Required to Be on Duty
8 A.M.–noon	4
Noon–4 P.M.	8
4 P.M.–8 P.M.	10
8 P.M.–midnight	6

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 parttime 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.

Formulate a linear programming model for this problem.

Let f₁ = number of full-time consultants working the morning shift (8 a.m.-4 p.m.),
f₂ = number of full-time consultants working the afternoon shift (Noon-8 p.m.),
f₃ = number of full-time consultants working the evening shift (4 p.m.-midnight),
p₁ = number of part-time consultants working the first shift (8 a.m.-noon),
p₂ = number of part-time consultants working the second shift (Noon-4 p.m.),
p₃ = number of part-time consultants working the third shift (4 p.m.-8 p.m.),
p₄ = number of part-time consultants working the fourth shift (8 p.m.-midnight).

(Click to select) maximize minimize C = (Click to select) (40 x 8) (30 x 8) (30 x 4) (40 x 4) (f₁ + f₂ + f₃) + (Click to select) (40 x 4) (40 x 8) (30 x 4) (30 x 8) (p₁ + p₂ + p₃ + p₄)

subject to (Click to select) f3 + p4 = 10 f1 + f2 + f3+ p3 = 8 f1 + p1 = 4 f1 + f2 = 4

(Click to select) f1 + p1 = 10 f1 + f2 + p2 = 8 f3 + p3 = 6 f1 + f3 + p3 = 8

(Click to select) f2 + f4 + p3 = 8 f2 + f3 + p3 = 10 f3 + f4 + p3 = 10 f2 + f3 + p3 = 8

(Click to select) f3 + f4 = 8 f3 + f4 = 6 f2 + f3 + p3 = 6 f3 + p4 = 6

f₁ = 2p₁

f₁ + f₂ = 2p₂

f₂ + f₃ = 2p₃

f₃ = 2p₄

f₁, f₂, f₃, p₁, p₂, p₃, p₄ = 0