08.07.2022 - 06:46

# 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 b

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 $18 per hour. The storage bin is open and operated 16 hours per day, 7 days per week, during the harvest season and is capable of unloading 35 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 30 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. Answers (1) • April 6, 2023 в 06:01 (a) The average number of trucks in the unloading system can be calculated using Little's Law, which states that the average number of entities in a system is equal to the average arrival rate multiplied by the average time spent in the system. In this case, the arrival rate is 30 trucks per hour, and the average time spent in the system can be calculated using the exponential distribution with a rate of 35 trucks per hour. Therefore, the average number of trucks in the unloading system is: Average number of trucks = Arrival rate x Average time in system Average number of trucks = 30 x (1/35) Average number of trucks = 0.857 trucks (b) The average time per truck in the system can also be calculated using Little's Law. The average time in the system is the inverse of the rate parameter in the exponential distribution, which is 1/35. Therefore, the average time per truck in the system is: Average time per truck = Average time in system / Average number of trucks Average time per truck = (1/35) / 0.857 Average time per truck = 0.027 hours or 1.6 minutes (c) The utilization rate for the bin area is the ratio of the arrival rate to the capacity of the system. In this case, the capacity of the system is 35 trucks per hour, so the utilization rate can be calculated as: Utilization rate = Arrival rate / Capacity Utilization rate = 30 / 35 Utilization rate = 0.857 or 85.7% (d) The probability that there are more than three trucks in the system at any given time can be calculated using the Poisson distribution. The parameter lambda, which represents the average number of arrivals per time period, is 30, and the cumulative distribution function for the Poisson distribution can be calculated as: P(X > 3) = 1 - P(X <= 3) P(X > 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)) P(X > 3) = 1 - (e^-30 * (1 + 30 + (30^2/2) + (30^3/6))) Therefore, the probability that there are more than three trucks in the system at any given time is \approx imately 0.725 or 72.5%. (e) The total daily cost to the farmers of having their trucks tied up in the unloading process can be calculated by multiplying the number of trucks in the system by the average time per truck in the system, and then multiplying by the cost per hour of$18. Since the system is operated for 16 hours per day, the total daily cost can be calculated as: Total daily cost = Average number of trucks x Average time per truck x Cost per hour x Operating hours per day Total daily cost = 0.857 x 0.027 x $18 x 16 Total daily cost =$7.37 per day Therefore, the total daily cost to the farmers of having their trucks tied up in the unloading process is \approx imately \$7.37 per day.