. Virtual Reality Goggle Inventory

a. Average Monthly Profit

To calculate the average monthly profit, we need to consider the following factors:

• Revenue from sales: Each pair of goggles sold generates a revenue of $300.

• Procurement cost: Each pair of goggles costs $150 to procure.

• Inventory holding cost: If there are any unsold goggles at the end of the month, Galaxy incurs an inventory holding cost of $20 per pair.

• Shortage cost: If Galaxy is unable to satisfy the entire demand for goggles, it incurs a shortage cost of $40 per unsatisfied unit.

Let’s denote the following variables:

• X = monthly demand for VR goggles (a normal random variable with mean 160 and standard deviation 40)

• I = initial inventory level (140 pairs)

• S = number of goggles sold

• R = revenue from sales

• H = inventory holding cost

• Sh = shortage cost

• P = monthly profit

The number of goggles sold depends on the demand and the initial inventory level:

S = min(X, I)

The revenue from sales is calculated as:

R = 300S

The inventory holding cost is incurred if there are any unsold goggles at the end of the month:

H = 20(I - S)

The shortage cost occurs if the demand exceeds the initial inventory level:

Sh = 40(max(0, X - I))

Finally, the monthly profit is calculated as:

P = R - H - Sh

To simulate the monthly profit, we can use a spreadsheet simulation model. Here’s an example of how to do this in Excel:

1. Create a table with the following columns:

   • Month
   • Demand (X)
   • Sold (S)
   • Revenue (R)
   • Holding Cost (H)
   • Shortage Cost (Sh)
   • Profit (P)

2. In the Demand column, generate 12 random numbers from a normal distribution with mean 160 and standard deviation 40.

3. In the Sold column, use the MIN function to calculate the minimum of the Demand and Initial Inventory columns.

4. In the Revenue column, calculate the product of the Sold and Selling Price columns.

5. In the Holding Cost column, calculate the product of the Unsold Goggles and Holding Cost per Goggle columns.

6. In the Shortage Cost column, calculate the product of the Unsatisfied Demand and Shortage Cost per Unit columns.

7. In the Profit column, calculate the difference between the Revenue, Holding Cost, and Shortage Cost columns.

8. Repeat steps 2-7 for 12 months to get a sample of 12 monthly profits.

9. Calculate the average of the 12 monthly profits to estimate the average monthly profit under the current policy.

b. Proportion of Months with Satisfied Demand

The proportion of months in which demand is completely satisfied is the proportion of months in which the demand is less than or equal to the initial inventory level.

Proportion of Satisfied Demand = P(X ≤ I)

Since X is a normal random variable with mean 160 and standard deviation 40, we can calculate this probability using the cumulative distribution function (CDF) of the normal distribution:

Proportion of Satisfied Demand = NORM.DIST(140, 160, 40, TRUE)

This formula returns approximately 0.7734, indicating that about 77.34% of the months will have satisfied demand.

c. Profitability Comparison

To compare the profitability of monthly replenishment levels of 140 and 160 pairs of goggles, we can run two separate simulations, one for each replenishment level. For each simulation, we can generate 12 monthly profits and calculate the average monthly profit.

Average Profit at 140 pairs = average(P_140)
Average Profit at 160 pairs = average(P_160)

We can then calculate the difference between the two average profits and construct a 95% confidence interval around this difference.

Difference = Average Profit at 160 pairs - Average Profit at 140 pairs

The 95% confidence interval can be calculated using the t-distribution since we have a sample of 12 observations for each replenishment level.

Confidence Interval = Difference ± t_0.975 *