Problem 1: Sampling Test Method for Minimizing Cost

To solve the problem of determining whether a batch of parts should be accepted or rejected, the company must develop a sampling method. This involves sequential testing with the goal of minimizing costs, where tests continue until the reliability requirements are met for accepting or rejecting the batch.

We will establish a mathematical model using probability to determine how many tests need to be performed iteratively, with the goal of achieving specific reliability thresholds based on the number of defective items found.

Definitions:

• Let $p_0$ be the nominal defective rate (maximum acceptable defective rate) of the parts, with $p_0 = 10% = 0.1$.
• $n$ represents the number of parts sampled.
• $k_n$ represents the number of defective parts found in $n$ samples.
• We aim to determine the number of samples $n$ such that:
  • Scenario (1): The batch will be rejected if the defective rate exceeds the nominal value $p_0$ with 95% reliability.
  • Scenario (2): The batch will be accepted if the defective rate does not exceed the nominal value $p_0$ with 90% reliability.

Assumptions:

1. The cost of testing individual parts is the same regardless of the number of parts tested.
2. The process is sequential, i.e., after each test, the result is evaluated to see if the reliability thresholds have been met.

Step-by-Step Approach

1. Binomial Distribution:

   • The number of defective parts in a sample of $n$ follows a binomial distribution, $X_n \sim \text{Binomial}(n, p)$, where $p$ is the defective rate.
   • The probability of observing $k_n$ defective parts in $n$ samples is: $$ P(X_n = k_n) = \binom{n}{k_n} p^{k_n} (1 - p)^{n - k_n} $$

2. Bayesian Update for Defective Rate:

   • After sampling $n$ parts and finding $k_n$ defective parts, we can update our estimate of the defective rate $p$ using Bayesian inference.
   • If the prior distribution of $p$ is uniform, the posterior distribution for $p$ is a Beta distribution: $$ p | k_n \sim \text{Beta}(k_n + 1, n - k_n + 1) $$
   • The mean of this posterior distribution is $\hat{p} = \frac{k_n + 1}{n + 2}$, which gives an estimate of the defective rate based on current data.

3. Reliability Thresholds:

   • To determine if the batch should be rejected or accepted, we need to calculate the cumulative probability of the defective rate being above or below the nominal value $p_0 = 0.1$.
   • This can be done using the Beta distribution's cumulative density function (CDF):
     • For rejection with 95% reliability: $$ P(p > p_0 | k_n, n) = 1 - F_{\text{Beta}}(p_0; k_n + 1, n - k_n + 1) \geq 0.95 $$ where $F_{\text{Beta}}$ is the CDF of the Beta distribution.
     • For acceptance with 90% reliability: $$ P(p \leq p_0 | k_n, n) = F_{\text{Beta}}(p_0; k_n + 1, n - k_n + 1) \geq 0.90 $$

4. Sampling Strategy:

   • The company starts testing parts one by one and keeps track of $n$ and $k_n$ (the number of defective parts found so far).
   • After each test, the company updates the posterior distribution of the defective rate and calculates the probabilities for rejection or acceptance.
   • Testing stops as soon as the probability meets the required reliability for either rejection (95%) or acceptance (90%).

Specific Results for the Two Scenarios

Let’s now implement the sampling test strategy for the two scenarios.

Scenario 1: Rejection with 95% Reliability

• The batch will be rejected if the probability that the defective rate exceeds 10% is greater than or equal to 95%.

Scenario 2: Acceptance with 90% Reliability

• The batch will be accepted if the probability that the defective rate is less than or equal to 10% is greater than or equal to 90%.

The number of tests $n$ required for each scenario depends on how many defective parts are found in the process. We will continue sampling parts until one of the conditions is satisfied.

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In practice, we would simulate the process to find the exact stopping points for both rejection and acceptance. The following algorithm can be used iteratively:

1. Start with $n = 1$.
2. After each test, update $k_n$ (number of defective parts) and calculate the posterior Beta distribution.
3. Check if the reliability thresholds are met:
   • If $P(p > p_0 | k_n, n) \geq 0.95$, reject the batch.
   • If $P(p \leq p_0 | k_n, n) \geq 0.90$, accept the batch.
4. If neither condition is met, continue testing.