The wheat harvesting season in the American Midwest is short, and most farmers d

Question

The wheat harvesting season in the American Midwest is short, and most farmers deliver their truck-loads 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 for a block 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, the cost of truck rental, and idle driver time are significant concerns to the cooperative members. Although farmers have difficulty quantifying crop damage, it is easy to assign a waiting and unloading cost for truck and driver of $ 17 per hour. During the 2-week harvest season,The storage bin is open and operated 16 hours per day, 7 days per week, during the harvest season and is capable of unloading 36 trucks per hour according to an exponential distribution. Full trucks arrive all day long (during the hours the bin is open) at a rate of about 31 per hour, following a Poisson pattern.

To help the cooperative get a handle on the problem of lost time while trucks are waiting in line or unloading at the bin, find the
(a) average number of trucks in the unloading system.
(b) average time per truck in the system.
(c) utilization rate for the bin area.
(d) probability that there are more than three trucks in the system at any given time.
(e) total daily cost to the farmers of having their trucks tied up in the unloading process.

F.) The cooperative, as mentioned, uses the storage bin only two weeks per year. Farmers estimate that enlarging the bin would cut unloading costs by 50% next year. It will cost $5,000 to do so during the off-season. Would it be worth the cooperative's while to enlarge the storage area?

Explanation / Answer

(a) average number of trucks in the unloading system, L =/(1-) = 6.2
(b) average time per truck in the system, W = L/ = 0.2 hours = 12 minutes
(c) utilization rate for the bin area, = 86.1%
(d) probability that there are more than three trucks in the system at any given time, 1 - (p0+p1+p2+p3) = 1-(13.89%+11.96%+10.3%+8.87%) = 54.98%. p0, p1, p2, p3 are calculated by the formula Pn =(1-)*^n, where n=0,1,2,3
(e) total daily cost to the farmers of having their trucks tied up in the unloading process = average time in system per truck*arrival rateper hour*16 hours per day*$17 per hour = 0.2 hours(as calculated in part b above) * 31*16*17 = $ 1686.4

(f) Total unloading costs during the season = $ 1686.4 * 7 days per week * 2 weeks of season = $ 23609.6 . Spending $ 5000 to enlarge the bin so as to save 50% of the unloading costs , i.e. $ 11804.8 in the next season is a proposal worth its while.

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