Problem 61 - Calculus and its applications

Question

Manufacturing. In an automotive body-welding line, delays encountered during the process can be modeled by various probability distributions. (Source: R. R. Inman, "Empirical Evaluation of Exponential and Independence Assumptions in Queueing Models of Manufacturing Systems," Production and Operations Management, Vol. \(8,409-432\) (1999).) The processing time for the robogate has a normal distribution with mean \(38.6 \mathrm{sec}\) and standard deviation 1.729 sec. Find the probability that the next operation of the robogate will take 40 sec or less.

Step-by-Step Solution

Answer

The probability is approximately 79.10%.

Step 1: Understand the Distribution

The processing time follows a normal distribution with a mean of \(\mu = 38.6\) seconds and a standard deviation of \(\sigma = 1.729\) seconds. We want to find the probability that the processing time is 40 seconds or less.

Step 2: Standardize the Value

We need to convert 40 seconds into a standard normal variable (a z-score), using the formula: \( z = \frac{x - \mu}{\sigma} \). Where \( x = 40 \), \( \mu = 38.6 \), and \( \sigma = 1.729 \).\[ z = \frac{40 - 38.6}{1.729} = \frac{1.4}{1.729} \approx 0.81 \]

Step 3: Use Z-Table or Normal Distribution Calculator

With \( z \approx 0.81 \), use a standard normal distribution table (or calculator) to find the probability of \( z \leq 0.81 \). The table or calculator will give you \( P(Z \leq 0.81) \).

Step 4: Interpret the Result

From the standard normal distribution table, \( P(Z \leq 0.81) \approx 0.7910 \). This means there is approximately a 79.10% probability that the next operation will take 40 seconds or less.

Key Concepts

Normal Distribution

The normal distribution, often referred to as a bell curve, is a key concept in statistics. It is widely used because many real-world phenomena tend to follow this pattern. The characteristics of a normal distribution include:

• Symmetrical shape about the mean.
• Mean, median, and mode are all equal.
• The curve is asymptotic, meaning it approaches the horizontal axis but never touches it.

In manufacturing processes like an automotive welding line, understanding this distribution helps in estimating the likelihood of certain events, such as the time it takes for an operation to complete. Knowing that processing times are normally distributed allows us to make predictions about future operations, aiding in process efficiency and resource planning.

Standard Deviation

Standard deviation is a measure of how spread out the numbers in a data set are. It indicates the average distance from the mean.

• A small standard deviation suggests that the data points tend to be close to the mean.
• A large standard deviation indicates that the data points are spread out over a wide range of values.

In the context of manufacturing, like the robogate operation, standard deviation helps us understand the variability in processing times. A lower standard deviation means the process is more consistent, which is vital for quality control and meeting production timelines.

Z-Score

A z-score provides a way to understand how far a specific data point is from the mean, measured in units of standard deviation.

• The formula to calculate the z-score is: \( z = \frac{x - \mu}{\sigma} \)
• A z-score of 0 indicates the value is exactly at the mean.
• Positive z-scores indicate values above the mean, while negative scores indicate values below the mean.

Using the z-score in manufacturing, we can determine the probability of certain processing times, by converting these times into a standard normal variable. This helps in anticipating delays and optimizing workflow.

Manufacturing Systems

In manufacturing systems, managing time efficiently directly impacts productivity and cost-effectiveness. A system like the automotive body-welding line must analyze various probability distributions to understand and minimize delays.

• Probability distributions, like the normal distribution, help model the time it takes for operations to complete.
• Consistent production processes rely on understanding and reducing variability, which the normal distribution analysis aids in.

By using statistical tools such as mean, standard deviation, and z-scores, manufacturing systems can ensure smoother operations and effectively anticipate any disruptions. This is crucial for optimizing processes, improving quality, and maintaining competitive advantage in the industry.