Go Bananas Breakfast Cereal At Great Grasslands Grains Inc

Go Bananas Breakfast Cereal Great Grasslands Grains Inc Ggg Manuf

Go Bananas! Breakfast cereal produced by Great Grasslands Grains Inc. involves careful quality control to ensure each 16-ounce box contains an appropriate amount of banana-flavored marshmallows. The company's management is concerned with maintaining a consistent product and has established a sample-based inspection policy, monitoring weekly samples of 25 boxes to detect any deviations from the standard. Specifically, they will halt production if five or more boxes in a weekly sample fail to meet the acceptable marshmallow weight range, which is between 1.6 ounces and 2.4 ounces. This policy hinges on the assumption that only 8% of all boxes fail to meet the standards when the process is under normal operation. This report addresses the implications of their current policy, explores how to adjust it to minimize unwarranted shutdowns, and considers the potential improvements if the production process is optimized.

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Introduction

Quality control is essential in food manufacturing to ensure product consistency, customer satisfaction, and compliance with safety standards. For Great Grasslands Grains Inc. (GGG), the focus is on maintaining an acceptable amount of banana-flavored marshmallows in their new cereal, Go Bananas!. Given the nature of the production process and the variability inherent in manufacturing, statistical sampling methods are employed to monitor product quality. The present policy involves analyzing a weekly sample of 25 boxes to determine if the production should be halted, based on the number of boxes that deviate from the standard. This paper discusses the probability of a shutdown under current procedures, recommends improvements to meet strict quality goals, and explores how process improvements could further enhance quality control.

Current Probability of Production Shutdown

The policy specifies that if five or more out of 25 sampled boxes fail the marshmallow weight standard, production is halted. Assuming the process is working properly, the failure rate (p) is 8% or 0.08. The number of failures in a sample follows a binomial distribution with parameters n=25 and p=0.08. The probability of observing at least five failures, which would trigger a shutdown, can be calculated as P(X ≥ 5).

Using the binomial probability model:

P(X ≥ 5) = 1 - P(X ≤ 4) = 1 - ∑_{k=0}^{4} C(25, k) p^k (1-p)^{25-k}

Calculating the cumulative probability P(X ≤ 4):

Using a calculator or statistical software, the sum of probabilities from k=0 to 4 can be obtained. The resulting value is approximately 0.893, which makes:

P(X ≥ 5) ≈ 1 - 0.893 = 0.107 or about 10.7%.

This indicates that, under normal operation, there is roughly a 10.7% chance that the sample will lead to a production shutdown. From a managerial perspective, this risk might be considered acceptable, or it might warrant adjustments depending on the company's tolerance for false alarms versus quality assurance.

Adjusting the Sample Fail Threshold to Minimize False Shutdowns

GGG management aspires to limit the probability of unnecessary shutdowns to no more than 1%, i.e., P(X ≥ r) ≤ 0.01, where r is the number of failed boxes in the sample that triggers shutdown. Given the current known failure rate p=0.08, the problem reduces to finding the largest r such that:

P(X ≥ r) ≤ 0.01

which is equivalent to:

P(X ≤ r - 1) ≥ 0.99.

To determine the appropriate r, we utilize the binomial cumulative distribution function (CDF) with n=25, p=0.08. Testing the values:

  • For r=1: P(X ≤ 0) = (1-p)^{25} ≈ 0.156 -> 15.6%
  • For r=2: P(X ≤ 1) = P(X=0) + P(X=1) = 0.156 + 25C(24,1)0.08*0.92^{24} ≈ 0.413 -> 41.3%
  • For r=3: P(X ≤ 2) ≈ 0.691 -> 69.1%
  • For r=4: P(X ≤ 3) ≈ 0.877 -> 87.7%
  • For r=5: P(X ≤ 4) ≈ 0.957 -> 95.7%
  • For r=6: P(X ≤ 5) ≈ 0.985 -> 98.5%

Since the goal is P(X ≥ r) ≤ 0.01, equivalently P(X ≤ r-1) ≥ 0.99, the r that satisfies this condition is r=6, because P(X ≤ 5) ≈ 98.5%. For P(X ≥ 6) = 1 - P(X ≤ 5) ≈ 1 - 0.985 = 0.015 or 1.5%, which is just above 1%. To meet the 1% threshold, the cutoff must be r=7, which corresponds to P(X ≤ 6). Using the cumulative probability, P(X ≤ 6) is approximately 0.997, thus P(X ≥ 7) ≈ 0.003, well below 1%. Therefore, setting the cutoff to 7 failures will limit false shutdowns to approximately 0.3%, satisfying management’s goal. This suggests that increasing the failure threshold from 5 to 6 or 7 can substantially reduce unwarranted production halts.

Reducing Failure Rate Through Process Improvement

Ms. Finkel's proposition to improve the process involves reducing the failure rate (p). To achieve a probability of at most 1% that five or more boxes fail in a sample, she must lower p such that P(X ≥ 5) in a binomial(n=25, p) distribution is less than or equal to 0.01.

Given the earlier calculations, when p=0.08, P(X ≥ 5) ≈ 10.7%, which is too high. To meet the target, p must be reduced significantly.

Let’s analyze the necessary failure probability p by reversing the calculations for the binomial distribution:

We target P(X ≥ 5) ≤ 0.01, which equals 1%. Equivalently, P(X ≤ 4) ≥ 0.99.

Now, perform approximate calculations to determine p:

  • Using binomial probability bounds and leveraging normal approximation for large n,
  • the mean μ = np, and the standard deviation σ = √(np*(1-p)).

To have P(X ≤ 4) ≥ 0.99, the expected number of failures μ must be sufficiently low, and the probability of observing 4 or fewer failures must be high. Applying the normal approximation with a continuity correction:

P(X ≤ 4) ≈ Φ[(4 + 0.5 - μ) / σ] ≥ 0.99, where Φ is the standard normal cumulative distribution function.

Given that for a standard normal Z-score corresponding to 99% probability is approximately 2.33, we have:

(4.5 - μ) / σ = 2.33

Substituting μ = 25p and σ = √(25p(1-p)):

(4.5 - 25p) / √(25p(1-p)) = 2.33

Simplifying:

(4.5 - 25p) ≈ 2.33 5 √(p(1-p)) = 11.65 * √(p(1-p))

Now, solving for p numerically involves iterative or algebraic methods, but a rough estimate suggests p should be around 0.02 or less for the probability to meet the criteria.

In conclusion, Ms. Finkel needs to reduce the failure rate from 8% to roughly 2% or lower to ensure that the probability of observing five or more failures in a sample of 25 boxes is 1% or less.

This substantial improvement necessitates significant process control enhancements, including better mixing, timing adjustments, or more precise ingredient measurement. The expected benefit is a decrease in false alarms, reducing unnecessary production halts and increasing overall efficiency and product quality.

Conclusion

GGG’s current quality control policy effectively balances the risk of undetected defects with false alarms, with initial probability estimates indicating a roughly 10.7% chance of shutting down production when it is working properly. However, management's desire to reduce false shutdowns to an absolute minimum, such as 1%, can be achieved by adjusting the failure threshold from 5 to 6 or 7 failed boxes in the weekly sample. Furthermore, substantial process improvements aiming to lower the failure rate from 8% to approximately 2% or less are critical for minimizing the risk of corrective actions and ensuring continuous production. Implementing robust quality control measures and process refinement is vital for maintaining product consistency and meeting stringent quality targets. These adjustments will help GGG maintain high standards, optimize production, and ensure customer satisfaction for their Go Bananas! cereal line.

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