Question

M6_IND2. The wheat harvesting system in the U.S. Midwest is short, and most farmers can only deliver truckloads of wheat to a giant central storage bin within a two-week span. Because of this, wheat filled trucks waiting to unload and return to the fields have been known to back up at the receiving bin. The central bin is owned cooperatively, and it is to every farmer’s benefit to make the unloading/storage process as efficient as possible. The cost of grain deterioration caused by unloading delays and the cost of the truck rental and idle driver time are significant concerns to the cooperative members. Although the farmers have difficulty quantifying crop damage, it is easy to assign a waiting and unloading cost for truck and driver of $23 per hour. The storage bin is open and operated 16 hours per day, seven days a week during the two-week harvesting season. The storage bin is capable of unloading 28 trucks per hour. Full trucks arrive all day long (during the hours that the storage bin is open) at a rate of 22 per hour. (Note: in this system there is not a separate server cost – the $23/hour combines the truck and driver. The cost would be based on the truck/driver waiting time only – no separate cost of a server) a) What is average number of trucks (to the nearest 0.1 trucks) in the unloading system? b) What is average time per truck (to the nearest 0.1 minutes) in the system? c) What is the utilization rate (to the nearest 0.01) for the bin area (enter as decimal - not percentage - i.e. enter 0.34 to represent 34% - do not enter 34)? d) What is the total daily cost (to the nearest $/day) to the farmers for having their trucks tied up in the unloading process? (Hint: this is not the same as the cost per truck per day) e) The cooperative, as mentioned, uses the storage bin only two weeks per year. So their annual costs are based on the costs during these two weeks. What is the annual waiting cost (to the nearest $) for the trucks during the unloading process (based on your total daily cost and the fact that they only deliver their loads during a two week span)? f) The farmers estimate that enlarging the bin would cut unloading costs by 50%. It would cost $10,000 to enlarge the bin. What would be the net savings (to the nearest $) be if the bin is enlarged? (Hint: What is the difference between the savings on the waiting costs and the cost of enlarging the bin – this can be a negative number) g) Based on your calculation of net savings - is it worth it to the cooperative to enlarge the storage area?

Answer 1

Answer :
Arrival rate, λ = 22 trucks per hour
Service rate, μ = 28 trucks per hour
Waiting and unloading cost, Cw = $ 23 per hour
a) Average number of trucks in the system, L = λ/(μ-λ) = 22/(28-22) = 3.7
b) Average time per truck in the system, W = 1/(μ-λ) = 1/(28-22) = 0.167 hours = 10.0 minutes
c) Utilization rate for the bin area = λ/μ = 22/28 = 0.79
d) Cost of waiting and unloading per hour = Cw*L = 3.7*23 = $ 84.33 per hour
Total daily cost = 84.33 * 16 hours per day = $ 1349.33 per day
e) Annual cost = $ 1349.33 per day * 2 weeks = 1349.33*14 = $ 18890.67 (assuming that the bin is operated for all 7 days and 16 hours every day for the two week span)
f) After 50% reduction, savings in unloading costs per year = 18890.67*50% = $ 9445.33
Net savings = 9445.33 - 10000 = - $ 554.67
g) Based on above computation, net saving is negative, therefore it is not worth to enlarge the storage area.