As the owner of Wildstyle Music you would like to develop an inventory ordering
policy which has at most a 10% probability of stocking out. Expected demand for
professional mixing headphones is 2190 per year. The store is open 365 days per year.
Every two weeks inventory is counted and a new order is placed. It takes 10 days for
the headphones to be delivered. Standard deviation of demand for the headphones
is 7 per week. There are currently 65 pairs of headphones in the store.
(a) (6 points) How many headphones should you order? (Hint: calculate the mean
and standard deviation of the daily demand first.)
To calculate the mean and standard deviation of the daily demand, we need to convert the weekly demand values to daily demand values:
Mean of weekly demand = 2190 / 52 weeks = 42.1154 headphones per week
Standard deviation of weekly demand = 7 headphones per week
Mean of daily demand = Mean of weekly demand / 7 days = 42.1154 / 7 = 6 headphones per day
Standard deviation of daily demand = Standard deviation of weekly demand / sqrt(7 days) = 7 / √7 ≈ 2.646 headphones per day
To calculate the number of headphones to order, we need to consider the lead time demand as well. The lead time is 10 days, and the expected daily demand during lead time is 6 headphones per day * 10 days = 60 headphones.
To achieve a maximum 10% probability of stocking out, we need to calculate the service level factor using the z-score corresponding to a 10% probability:
Z-score = invNorm(1 - 0.10) = invNorm(0.90) ≈ 1.28
Service level factor = 1 + Z-score = 1 + 1.28 = 2.28
To calculate the safety stock, we multiply the standard deviation of daily demand by the service level factor:
Safety stock = Standard deviation of daily demand * Service level factor ≈ 2.646 * 2.28 ≈ 6.03
Optimal inventory level = Average daily demand + Safety stock = 6 + 6.03 = 12.03
Since there are currently 65 pairs of headphones in the store, we subtract the current inventory level from the optimal inventory level to determine the number of headphones to order:
Number of headphones to order = Optimal inventory level - Current inventory level ≈ 12.03 - 65 ≈ -52.97
Since we cannot order a negative quantity, we should order 0 headphones.
To determine how many headphones you should order, we need to calculate the mean and standard deviation of the daily demand first.
Given:
- Expected demand for professional mixing headphones per year: 2190
- Store is open 365 days per year
- Inventory is counted every two weeks (14 days) and new order is placed
- It takes 10 days for the headphones to be delivered
- Standard deviation of demand for headphones is 7 per week
- Currently, there are 65 pairs of headphones in the store
First, let's calculate the mean and standard deviation of daily demand:
Mean daily demand = (Expected annual demand) / (Number of days per year)
Mean daily demand = 2190 / 365
Mean daily demand ≈ 6 headphones
Next, we need to calculate the standard deviation of daily demand. Since the standard deviation given is per week, we need to adjust it for the number of days per week:
Standard deviation of daily demand = (Standard deviation of weekly demand) / sqrt(Number of days per week)
Standard deviation of daily demand = 7 / sqrt(7) ≈ 2.646 headphones
Now let's calculate the number of headphones needed to maintain a 10% probability of stocking out:
Safety stock = (Z-score for desired service level) * (Standard deviation of demand during lead time)
Z-score for a 10% probability of stocking out (from standard normal distribution table) ≈ 1.28
Standard deviation of demand during lead time = (Standard deviation of daily demand) * sqrt(Lead time)
Since the inventory is counted every two weeks (14 days) and it takes 10 days for the headphones to be delivered:
Standard deviation of demand during lead time = (Standard deviation of daily demand) * sqrt(14 + 10)
Standard deviation of demand during lead time ≈ 2.646 * sqrt(24) ≈ 13.605 headphones
Safety stock = 1.28 * 13.605 ≈ 17.401 headphones
Now, we can calculate the number of headphones needed to be ordered:
Order Quantity = (Mean daily demand * Lead time) + Safety stock - Inventory on hand
Lead time = Inventory count period + Delivery time = 14 + 10 = 24 days
Order Quantity = (6 * 24) + 17.401 - 65 ≈ 131 headphones
Therefore, you should order approximately 131 headphones to maintain a 10% probability of stocking out.
a) D = 2190 per year,
μd = Daily Demand = 2190/365 = 6 pairs per day,
T = time between orders (review) = 14 days,
L = Lead time = 10 days
σd=Standard deviation of daily demand = σw/ 2.64 per day,
σLD+T= .I = Current Inventory = 65 pairs,
Service Level P = 90%