[Q 3-4] JuiceIt sells cold-pressed, organic juice at a local farmer’s market every weekend. To offer fresh juice, they make their famous “Energy Boost” on Saturday morning. Each bottle of “Energy Boost” sells for $8.00 and costs $4.50 to make. Any bottles of “Energy Boost” left unsold on Saturday are re-stocked at a cost of $1.75 each and are sold the next day at a discounted price of $5.00. The demand for “Energy Boost” is Poisson distributed with a mean of 9.
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Q3 What is the optimum quantity of “Energy Boost” bottles JuiceIt should make on Saturday morning?
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Q 4 Suppose JuiceIt decides to make and stock just enough bottles of “Energy Boost” so that the stock-out probability on Saturday is at most 5%.
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How many bottles of “Energy Boost” bottles should JuiceIt make on Saturday morning?
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Let's address both questions step by step.

### Q3: Optimal Quantity of "Energy Boost" Bottles for Saturday Morning

To determine the optimal quantity of "Energy Boost" bottles JuiceIt should make, we need to consider the balance between the cost of overstocking (making too many bottles) and understocking (making too few bottles). 

This is a typical **Newsvendor problem**. The goal is to minimize the expected total cost, and the optimal order quantity ($Q^*$) is found using the following formula:

\[
Q^* = F^{-1} \left( \frac{c_u}{c_u + c_o} \right)
\]

Where:
- $F^{-1}$ is the inverse of the cumulative distribution function (CDF) of the demand distribution (Poisson in this case).
- $c_u$ is the underage cost, or the cost of understocking (i.e., the lost profit when a bottle is not available for sale).
- $c_o$ is the overage cost, or the cost of overstocking (i.e., the extra cost when bottles are left unsold).

#### Underage Cost ($c_u$):

This is the opportunity cost of not having enough bottles to meet demand. Each bottle that could have been sold on Saturday but wasn't is a loss of potential profit:

\[
c_u = \text{Selling Price on Saturday} - \text{Cost to Make}
\]
\[
c_u = 8.00 - 4.50 = 3.50
\]

#### Overage Cost ($c_o$):

This is the extra cost of having leftover bottles that must be re-stocked on Sunday. If a bottle is leftover, it incurs the re-stocking cost and then sells at a discounted price on Sunday. Thus, the overage cost is:

\[
c_o = \text{Cost to Make} - (\text{Discounted Selling Price} - \text{Re-stocking Cost})
\]
\[
c_o = 4.50 - (5.00 - 1.75) = 4.50 - 3.25 = 1.25
\]

#### Critical Ratio:
The critical ratio is the ratio of the underage cost to the sum of the underage and overage costs:

\[
\frac{c_u}{c_u + c_o} = \frac{3.50}{3.50 + 1.25} = \frac{3.50}{4.75} = 0.7368
\]

We now need to find the Poisson demand that corresponds to a cumulative probability of 0.7368.

Since the demand is Poisson-distributed with a mean of 9, we calculate the cumulative probabilities of different demand levels to find the quantity corresponding to 0.7368.

### Q4: Stock-Out Probability of at Most 5%

In this case, JuiceIt wants the probability of running out of stock (demand exceeding supply) to be at most 5%. This means the company wants a **95% service level** (i.e., they want to meet the demand 95% of the time).

The service level is the cumulative probability of the demand being less than or equal to the stock level. Therefore, we need to find the Poisson demand level that corresponds to a cumulative probability of 95%.

This can also be done by finding the inverse CDF of the Poisson distribution with a mean of 9 that corresponds to a probability of 0.95.

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Let me now compute the specific demand values for both cases.### Results:

- **Q3 (Optimal Quantity)**: JuiceIt should make **11 bottles** of "Energy Boost" on Saturday morning to minimize the total expected costs.
  
- **Q4 (Stock-Out Probability ≤ 5%)**: JuiceIt should make **14 bottles** to ensure that the probability of running out of stock on Saturday is at most 5%.

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Would you like any further details or explanations on this solution?

Here are 5 related questions that might interest you:
1. How does the Poisson distribution affect demand forecasting?
2. What happens if the cost to make each bottle increases? How does that change the optimum?
3. Can we apply a different distribution for demand if data suggests another pattern?
4. How do changes in restocking costs impact the overage and underage costs?
5. What other inventory optimization models could JuiceIt use in this scenario?

**Tip**: The critical ratio method balances the costs of overstocking and understocking, which is vital for determining inventory levels when demand is uncertain.

JuiceIt sells cold-pressed, organic juice at a local farmer’s market every weekend. Each bottle of 'Energy Boost' sells for $8.00 and costs $4.50 to make. Leftover bottles are re-stocked at a cost of $1.75 and sold at a discounted price of $5.00 on Sunday. The demand for 'Energy Boost' is Poisson distributed with a mean of 9. What is the optimum quantity of bottles JuiceIt should make on Saturday morning? Also, how many bottles should JuiceIt make if the stock-out probability on Saturday is at most 5%?

Determine the optimal inventory for JuiceIt's 'Energy Boost' bottles using Poisson-distributed demand and the Newsvendor model. Learn how to balance overage and underage costs to minimize total expected costs and how to meet a 95% service level. Includes detailed cost breakdown and Poisson distribution calculations.

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