Problem 31 - Applied Statistics and Probability for Engineers

Question

An article in Clinical Infectious Diseases ["Strengthening the Supply of Routinely Administered Vaccines in the United States: Problems and Proposed Solutions" (2006, Vol.42(3), pp. S97-S103)] reported that recommended vaccines for infants and children were periodically unavailable or in short supply in the United States. Although the number of doses demanded each month is a discrete random variable, the large demands can be approximated with a continuous probability distribution. Suppose that the monthly demands for two of those vaccines, namely measles-mumps-rubella (MMR) and varicella (for chickenpox), are independently, normally distributed with means of 1.1 and 0.55 million doses and standard deviations of 0.3 and 0.1 million doses, respectively. Also suppose that the inventory levels at the beginning of a given month for MMR and varicella vaccines are 1.2 and 0.6 million doses, respectively. (a) What is the probability that there is no shortage of either vaccine in a month without any vaccine production? (b) To what should inventory levels be set so that the probability is \(90 \%\) that there is no shortage of either vaccine in a month without production? Can there be more than one answer? Explain.

Step-by-Step Solution

Answer

(a) Probability is 0.435, (b) Higher inventory levels, multiple setups possible.

Step 1: Identify Random Variables and Distributions

The problem states that the monthly demands for MMR and varicella vaccines are independently, normally distributed. We denote the demand for MMR and varicella as \( D_{\text{MMR}} \) and \( D_{\text{Varicella}} \), respectively. \( D_{\text{MMR}} \sim \mathcal{N}(1.1, 0.3^2) \) and \( D_{\text{Varicella}} \sim \mathcal{N}(0.55, 0.1^2) \).

Step 2: Define Conditions for No Shortage

For there to be no shortage, the demand must not exceed the inventory. Therefore, for MMR: \( D_{\text{MMR}} \leq 1.2 \ million \) doses and for varicella: \( D_{\text{Varicella}} \leq 0.6 \ million \) doses.

Step 3: Calculate Probability Using Standard Normal Distribution

We calculate the probability \( P(D_{\text{MMR}} \leq 1.2) \) and \( P(D_{\text{Varicella}} \leq 0.6) \) assuming normal distribution. We find the z-scores: \[ z_{\text{MMR}} = \frac{1.2 - 1.1}{0.3} = \frac{0.1}{0.3} \approx 0.333 \]\[ z_{\text{Varicella}} = \frac{0.6 - 0.55}{0.1} = \frac{0.05}{0.1} = 0.5 \]Using a standard normal distribution table, find corresponding probabilities: \[ P(Z \leq 0.333) \approx 0.6293 \]\[ P(Z \leq 0.5) \approx 0.6915 \]

Step 4: Calculate Joint Probability for No Shortage

Since the demands are independent, the joint probability of no shortage is the product of the individual probabilities:\[ P(D_{\text{MMR}} \leq 1.2) \times P(D_{\text{Varicella}} \leq 0.6) = 0.6293 \times 0.6915 \approx 0.435 \]

Step 5: Determine Required Inventory Levels for 90% Probability

Set up the condition that \(P(D_{\text{MMR}} \leq x_{\text{MMR}}) \times P(D_{\text{Varicella}} \leq x_{\text{Varicella}}) = 0.9\). Assume each condition independently achieves approximately 95% to find inventory levels. For MMR, find \(x_{\text{MMR}}\) where \(P(Z \leq z) = 0.95\):\[ z = 1.645 \rightarrow x_{\text{MMR}} = 1.1 + 1.645 \times 0.3 \approx 1.5935\]For varicella, find \(x_{\text{Varicella}}\) where \(P(Z \leq z) = 0.95\):\[ z = 1.645 \rightarrow x_{\text{Varicella}} = 0.55 + 1.645 \times 0.1 \approx 0.7145\]Rounding these values usually due to inventory constraints or other conditions could lead to multiple satisfactory answers, but typically exact calculations are preffered in planning.

Key Concepts

Normal Distribution

The normal distribution, often known as the bell curve, is a fundamental concept in probability and statistics. It's used to describe data that clusters around a mean or average. In this exercise, the monthly demands for the MMR and varicella vaccines are modeled using normal distributions. Understanding this distribution is crucial for predicting whether vaccine supply will meet demand.

Characteristics of a normal distribution include:

• The curve is symmetrical around the mean, meaning data is evenly distributed from the center.
• Most values lie within three standard deviations from the mean.
• The area under the curve represents the total probability, which equals 1, or 100%.

The normal distribution helps determine the likelihood of certain demand levels and guides inventory management decisions, ensuring sufficient vaccine supply without overstocking.

Inventory Management

Inventory management in healthcare involves ensuring that the supply of vaccines meets the demand without significant shortages or excess. It requires balancing delicate factors such as unpredictable demand and the perishable nature of vaccines. This exercise highlights the importance of setting adequate inventory levels.

Key considerations include:

• Understanding demand patterns through historical data to predict future needs.
• Setting inventory levels based on demand forecasts and acceptable risk levels of shortage.
• Regularly assessing inventory to adjust for unexpected changes in demand.

Effective inventory management is essential as it minimizes costs and ensures that healthcare providers have the necessary vaccines to protect community health.

Joint Probability

Joint probability calculates the likelihood of two events occurring simultaneously. In this context, it relates to the probability that both MMR and varicella vaccines have no shortages in the same month. The formula for independent events is straightforward: multiply the probability of each event.

This step is crucial in managing multiple vaccine inventories to avoid shortages:

• Assess probabilities individually for each vaccine's demand not exceeding the inventory.
• Multiply these probabilities to find the joint probability of no shortage for both.
• Understand how dependencies between events could change calculations, although here it is assumed they are independent.

Joint probabilities help healthcare managers devise strategies to cover simultaneous demands and avoid crises.

Z-score

The Z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It helps standardize scores on different scales and is crucial in calculating normal probabilities. In this exercise, the Z-score calculation allows us to determine the likelihood of demand falling within a certain level of inventory.

Here’s how it works:

• Calculate the Z-score using the formula: \( z = \frac{(x - \mu)}{\sigma} \) where \( x \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
• Use Z-scores to find the probability that demand will not exceed inventory levels.
• The standard normal distribution table is used to find probabilities corresponding to specific Z-scores.

Z-scores are indispensable for decision-making in inventory management, providing insight into the probability and adequacy of current stock levels.