D.12 The wheat harvesting season in the Canadian Prairies is short, and farmers

Question

D.12 The wheat harvesting season in the Canadian Prairies is short, and farmers deliver their 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 for a block at the receiving bin. The central bin is owned tively, and it is to every farmer's benefit to make the unload- ing/storage process as efficient as possible. The cost of grain deteri- ation caused by unloading delays and the cost of truck rental and idle driver time are significant concerns to the co-operative mem- bers. Although farmers have difficulty quantifying crop damage, it or

Explanation / Answer

A = Rate of arrival = 30 per hour
S = Rate of service = 35 per hour

Since there is only one storage bin, we take the number of server =1 and thus model this as an M/M/1 queuing system.

Ls = Average number of trucks in the unloading system = A / (S - A) = 30/(35 - 30) = 6 trucks ---(a)

Ws = Average time a truck spends in the system = Ls / A = 6/30 hours = 12 minutes ---(b)

u = Utilization rate = A / S = 30/35 = 85.71% ---(c)

Pn>3 = Probability that there are more than 3 trucks in the system = u3+1 = (30/35)4 = 0.54 ---(d)

Hourly cost of waiting = $18 x Ls = $108
So, daily cost of waiting = $108 x 16 = $1,728 ---(e)

With unloading cost being 50%, the daily cost of waiting becomes = $9 x Ls x 16 = 9 x 6 x 16 = $864
So, the total cost of wating for the two weeks = $864 x 14 = $12,096
Since the investment of $9,000 is less than the cost saved ($12,096), enlarging the bin is beneficial ---(f)

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