Question

A receiving operator for a large grocery store is analyzing her operations. Trucks arrive to the loading dock at an average rate of four per hour for each day. The cost of operating a truck is estimated to be $75 per hour. Trucks are met by a two-person crew, the crew can unload the truck in an average of 9 minutes. The payroll associated with each crew member is $18/hour. It is possible to install new equipment to help the crew operate more efficiently, decreasing the unloading time from 9 minutes to 7 minutes per truck. Rental of this equipment would increase the daily cost of the operation by $200 per day. Assume each day is an 8-hour shift. Should the new equipment be installed?

a) Consider the performance of the crew before the new equipment is installed. On average, how many trucks are in the system (to the nearest 0.01 trucks) given the arrival and service rates?

b) Consider the performance of the crew before the new equipment is installed. On average, how many hours (to the nearest 0.01 hours) does each truck spend in the system given the arrival and service rates?

c) Consider the performance of the crew before the new equipment is installed. Basing your cost on the number of trucks in the system, what is the total system cost (to the nearest dollar) per day (crew cost and truck cost)?

d) Consider the performance of the crew after the new equipment is installed. On average, how many trucks (to the nearest 0.01 trucks) are in the system given the arrival and the new service rate?

e) Consider the performance of the crew after the new equipment is installed. On average, how many hours (to the nearest 0.01 hours) does each truck spend in the system given the arrival and the new service rate?

f) Consider the performance of the crew after the new equipment is installed. Basing your cost on the number of trucks in the system, what is the total system cost (to the nearest dollar) per day (crew cost and truck cost)? g) Based on your cost analysis - is it worth it to install the new equipment?

Answer 1

Arrival rate, λ = 4 per hour

Service rate = 1/service time = (1 / 9 minute) * 60 minutes per hour = 20/3 = 6.67 per hour

a) Average number of trucks in system, L = λ/(μ-λ) = 4 / (20/3-4) = 1.5 trucks

b) Average time spent in the system, W = L/λ = 1.5/4 = 0.375 = 0.38 hour

c) Total system cost per day = (2*18 + 75*1.5) *8 = $ 1188 per hour

d) New service rate, μ' = (1/7 minutes) *60 minutes per hour = 8.57 per hour

Average number of trucks in system, L = λ/(μ'-λ) = 4/ (60/7 - 4) = 0.875 = 0.88 trucks

e) Average time spent in the system, W = L/λ = 0.875/4 = 0.2187 = 0.22 hour

f) Total system cost per day = (2*18 + 90*0.88) *8 + 200 = $ 1121.6 = 1122 per day

g) Based on above cost analysis, we see that Total system cost per day is lesser in after the new equipment is installed. Therefore, it is worth to install the new equipment.

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