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In analyzing the wheat harvesting and unloading process at the cooperative’s central bin, the primary goal is to evaluate the efficiency of the current system using queuing theory, specifically the M/M/1 queuing model, which is suitable given the exponential distribution of service times and Poisson arrivals. The problem presents data on truck arrival rates, service rates, operational hours, and associated costs, enabling computation of performance metrics such as average number of trucks in the system, waiting times, utilization rates, probability distributions, and economic implications of system changes, such as enlarging the storage bin.

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Understanding the operational efficiency of grain unloading systems is vital for optimizing costs and minimizing delays that lead to crop deterioration and increased labor expenses. The scenario provided involves truck arrivals following a Poisson process at an average rate of 30 trucks per hour over a 16-hour workday, with the unloading process modeled as an exponential service time with a capacity to service 35 trucks per hour. This configuration aligns with an M/M/1 queuing system, characterized by a single server and exponential inter-arrival and service times. The analysis of this system reveals key performance indicators that influence operational decisions and strategic investments.

System Utilization and Average Number of Trucks in the System

The first metric to evaluate is the utilization rate of the unloading bin, computed as the ratio of the arrival rate λ to the service rate μ:

U = λ / μ = 30 / 35 ≈ 0.8571 or 85.71%

This indicates that the system is busy most of the time but still within capacity, meaning it almost reaches saturation but maintains manageable queue lengths. The average number of trucks in the system, L, includes both those being unloaded and those waiting in line, given by:

L = λ / (μ - λ) = 30 / (35 - 30) = 30 / 5 = 6 trucks

This average reflects that, at any given time during the busy harvest days, about six trucks are either unloading or waiting.

Average Time in the System and Waiting Time

The average time each truck spends in the system, W, encompasses waiting and unloading time:

W = 1 / (μ - λ) ≈ 1 / 5 = 0.2 hours or 12 minutes

Similarly, the average waiting time in the queue, Wq, is:

Wq = λ / (μ (μ - λ)) ≈ 30 / (35 * 5) ≈ 0.1714 hours or approximately 10.28 minutes

This indicates that trucks typically wait around 10 minutes before unloading begins, which is critical for assessing delays and potential costs.

Queue Length and Probability of Queue Sizes

The average number of trucks waiting in line, Lq, is derived as:

Lq = λ^2 / (μ (μ - λ)) ≈ 900 / (35 * 5) ≈ 5.14 trucks

The probability that the system is empty, P0, is calculated as:

P0 = 1 - U ≈ 1 - 0.8571 ≈ 0.1429 or 14.29%

Furthermore, the probability that there are more than three trucks in the system is computed based on the Poisson distribution:

• P(n > 3) = 1 - P(n ≤ 3)
• Calculated as the sum of probabilities from 0 to 3 trucks in the system, subtracted from one, gives approximately 54%.

Economic Implications and Cost Analysis

The cost associated with truck waiting times is estimated at $18 per hour per truck. To determine the total daily cost, multiply the average number of trucks waiting and unloading by the waiting time and cost per hour. The total number of trucks per day is 30 per hour over 16 hours, totaling 480 trucks daily.

Using the average waiting time per truck (about 0.2 hours), the total daily cost is:

Total cost ≈ Number of trucks average waiting time cost per hour = 480 0.2 18 = $1,728 per day

This significant figure underscores the importance of reducing delays, which could be achieved through infrastructure improvements.

Evaluating the Effectiveness of Expanding the Storage Bin

Farmers suggest enlarging the storage bin to cut unloading costs by 50%. The potential benefit comes from decreasing the queuing delay, thereby reducing idle driver time and crop deterioration costs. The analysis considers whether the investment in enlarging storage capacity—estimated to reduce unloading costs—would be justified economically.

Assuming enlarging the bin reduces the unloading cost by half, and this expansion would be utilized only during the two-week harvest window, the annual savings can be calculated by considering the reduction in total waiting time and associated costs. The critical financial analysis involves comparing the cost of expansion (capital expenditure) with the savings on unloading delays.

If the current cost per day due to delays is approximately $1,728, and an expansion could halve this to about $864, then the savings over only two weeks (14 days) amount to:

Savings = (1,728 - 864) * 14 ≈ $12,096

If the cost of enlarging the bin is less than this amount, it is economically viable. Additionally, the reduction in queue length and waiting time also reduces crop spoilage and associated losses.

Conclusion

Applying queuing theory to this scenario demonstrates that the current system operates near capacity with significant waiting times and costs. There is a clear case for infrastructure investment, especially enlarging storage capacity, which could substantially diminish operational costs and delays during the critical harvest period. Such analysis supports strategic planning for capital improvements that optimize throughput and minimize losses, hence increasing overall efficiency and profitability for the cooperative farmers.

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