Trucks arrive at the loading dock of a wholesale grocer at the rate of 1.2 per h
ID: 340410 • Letter: T
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 $60. The manager is thinking of adding another member to the crew. The service rate would then be 2.4 trucks per hour. Assume rates are Poisson.
Would the third crew member be economical? (Do not round intermediate calculations. Omit the "$" sign in your response.)
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.)
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.
Explanation / Answer
Considering ? as arrival rate, ? as service rate, Lq as the average number of members waiting for service, Ls 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 = Ls= Lq + 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 = Ls= Lq + 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 = Ls= Lq + 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).
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