Question

Trucks arrive at the loading dock of a wholesale grocer at the rate of 1.2 per hour. A single crew consisting of two workers can load a truck in about 30 minutes. Crew members receive $10 per hour in wages and fringe benefits, and trucks and drivers reflect an hourly cost of $60. The manager is thinking of adding another member to the crew. The service rate would then be 2.4 trucks per hour. Assume rates are Poisson.

a.	Would the third crew member be economical? (Do not round intermediate calculations. Omit the "$" sign in your response.)

(Click to select)YesNo because, the total cost is $ _______.

b.	Would a fourth member be justifiable if the resulting service capacity were 2.6 trucks per hour? (Do not round intermediate calculations. Round your answer to 2 decimal places. Omit the "$" sign in your response.)

(Click to select)YesNo because, the total cost is $ ________.

Answer 1

Considering ? as arrival rate, ? as service rate, L_q as the average number of members waiting for service, L_s as the average number of the members in the system, r as the average number of members being served.

Given that , Trucks arrive at the loading dock of a wholesale grocer, ?= 1.2 per hour

Case 1: Single crew has two workers, ?= 2 per hour

Therefore, the average number of members = L_s= L_q + r = ?²/(?*(?- ?)) + (?/?) = [(1.2)*(1.2) / (2* (2-1.2))] + (1.2/2) = 0.9 + 0.6 = 1.5 trucks.

Total cost = Waiting cost + Capacity cost

Therefore, total cost with 2 members crew = 2*$10 + 1.5*$60 = $110 per hour

Case 2: When manager introduces another crew member hence service rate = ?= 2.4 per hour

Therefore, the average number of members = L_s= L_q + r = ?²/(?*(?- ?)) + (?/?) =[1.2*1.2] / (2.4 * (2.4 – 1.2))] + (1.2 / 2.4) = 0.5 + 0.5 = 1 truck.

Hence, total cost with 3 members crew = 3*$10 + 1*$60 = $90 per hour.

Answer (a) : Yes, the third crew member be economical with the total cost of $90 per hour.

Case 3: introducing one more member that means crew of 4 members, with the service rate = ?= 2.6 per hour,

Therefore, the average number of members = L_s= L_q + r = ?²/(?*(?- ?)) + (?/?) =[1.2*1.2] / (2.6 *(2.6 – 1.2))] + (1.2 / 2.6)= 0.396 + 0.461   = 0.857 person.

Therefore, total cost with 4 members crew = 4 * $10 + 0.857 * $60 = $91.42 per hour.

Answer (b) : No, a fourth member may not be justifiable as the total cost ($91.42 per hour) is more compared to three member crew ($90 per hour).