An assembly line produces 800 products per hour, with a standard deviation of demand during the lead time of 100. For every product they make, they use a Kanban system. The lead time from the inventory location to the assembly line is 2 hours. The boxes, the product is transported in can carry 200 items, and the company uses an Alfa factor of 0.1.

3a. What’s the number of Kanban cards they need?

3b. If they instead want to use a ROP system, what will the ROP be?

3c. What service level does the safety stock represent? (The probability of not having a shortage during one inventory cycle.)

Let's break down the problem and solve each part step by step.

### Part 3a: Number of Kanban Cards
To calculate the number of Kanban cards needed, we use the formula:

\[
\text{Number of Kanban Cards} = \frac{D \times L (1 + \alpha)}{C}
\]

Where:
- \(D\) = Demand rate per hour
- \(L\) = Lead time (in hours)
- \(\alpha\) = Safety factor (Alfa factor)
- \(C\) = Container capacity

Given:
- \(D = 800 \, \text{products/hour}\)
- \(L = 2 \, \text{hours}\)
- \(\alpha = 0.1\)
- \(C = 200 \, \text{items}\)

Substituting the values:

\[
\text{Number of Kanban Cards} = \frac{800 \times 2 \times (1 + 0.1)}{200} = \frac{800 \times 2 \times 1.1}{200} = \frac{1760}{200} = 8.8
\]

Since the number of Kanban cards must be an integer, they need **9 Kanban cards**.

### Part 3b: Reorder Point (ROP) System
The Reorder Point (ROP) is given by:

\[
\text{ROP} = D \times L + \text{Safety Stock}
\]

Safety stock is calculated as:

\[
\text{Safety Stock} = Z \times \sigma_{LT}
\]

Where:
- \(Z\) is the Z-score corresponding to the desired service level
- \(\sigma_{LT}\) is the standard deviation during lead time

First, we calculate the demand during lead time:

\[
\text{Demand during lead time} = 800 \times 2 = 1600 \, \text{products}
\]

Since the safety stock is not provided directly, we'll keep it general until we solve Part 3c. The ROP formula remains:

\[
\text{ROP} = 1600 + Z \times 100
\]

### Part 3c: Service Level Represented by Safety Stock
To find the service level, we use the inverse relationship between the Z-score and the probability of stockout (service level):

\[
\text{Service Level} = \Phi(Z)
\]

Where \( \Phi(Z) \) is the cumulative distribution function (CDF) of the standard normal distribution.

Given that the Alfa factor \( \alpha = 0.1 \), the safety stock is based on:

\[
\text{Safety Stock} = 0.1 \times D \times L = 0.1 \times 1600 = 160 \, \text{products}
\]

To find the Z-score:

\[
Z = \frac{\text{Safety Stock}}{\sigma_{LT}} = \frac{160}{100} = 1.6
\]

Using a standard normal distribution table, a Z-score of 1.6 corresponds to a service level of approximately **0.9452**, or **94.52%**.

### Summary:
- **3a.** The number of Kanban cards needed is **9**.
- **3b.** The Reorder Point (ROP) is **\(1600 + 100Z\)** products.
- **3c.** The safety stock represents a service level of **94.52%**.

Would you like further details on any part of the solution?

### Relative Questions
1. How does changing the lead time affect the number of Kanban cards required?
2. What would happen to the ROP if the standard deviation of demand increased?
3. How can the Alfa factor influence inventory management?
4. What Z-score corresponds to a 99% service level?
5. If the container size changed to 250 items, how would the Kanban calculation change?
6. How does increasing the demand rate impact the Kanban cards and ROP?
7. Can a company use the Kanban and ROP systems together effectively?
8. How can companies reduce the lead time to improve efficiency?

### Tip:
Understanding how the demand variability (standard deviation) impacts inventory levels can help optimize stock and reduce costs.

Explore inventory management using Kanban and Reorder Point (ROP) systems. Calculate the number of Kanban cards needed, determine the ROP, and understand the service level represented by safety stock in this detailed solution.

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