operations management •• 12.51 Cynthia Knott’s oyster bar buys fresh Louisiana oysters for $5 per pound and sells them for $9 per pound. Any oysters not sold that day are sold to her cousin, who has a nearby grocery store, for $2 per pound. Cynthia believes that demand follows the normal distribution, with a mean of 100 pounds and a standard deviation of 15 pounds. How many pounds should she order each day?
If she buys x oysters and has y left over, then her profit is
9(x-y)-5x+2y = 4x-7y
So, if her goal is to break even, she needs y = 4/7 x
That is, she needs to sell at least 3/7 of he oysters.
If she has some other goal, then that must be adjusted.
Now you can play around with the Z-table, and a good place is
http://davidmlane.com/hyperstat/z_table.html
To determine how many pounds Cynthia should order each day, we need to consider the following factors:
1. The expected demand for oysters each day.
2. The probability of running out of oysters each day (stockout probability).
3. The cost of ordering too many oysters (excess inventory cost).
Let's break down the calculation step-by-step:
Step 1: Calculate the expected demand
The mean demand is given as 100 pounds. However, since demand follows a normal distribution, we need to consider the mean and standard deviation. The expected demand can be calculated by multiplying the mean demand by a factor known as the "z-value" for a given service level. A common service level is 95%, so the z-value for a 95% service level is approximately 1.645.
Expected demand = Mean demand + (Z-value * Standard deviation)
Expected demand = 100 + (1.645 * 15)
Expected demand = 124.68 pounds (approx.)
Step 2: Calculate the stockout probability
The stockout probability is the probability of not being able to meet the demand. It can be calculated using the standard normal distribution table or statistical software. For a given service level (e.g., 95%), the stockout probability is the area under the curve to the right of the expected demand.
Stockout probability = 1 - Service level
Stockout probability = 1 - 0.95
Stockout probability = 0.05
Step 3: Calculate the maximum acceptable shortage quantity
The maximum acceptable shortage quantity is the amount of oysters Cynthia is willing to run out of each day (in pounds). This is subjective and depends on the specific situation. Let's assume Cynthia is willing to run out of a maximum of 10 pounds of oysters each day.
Maximum acceptable shortage quantity = Maximum shortage quantity + Average leftover quantity
Maximum acceptable shortage quantity = 10 pounds
Step 4: Calculate the ordering quantity
The ordering quantity can be calculated using the following formula:
Ordering quantity = Expected demand + Maximum acceptable shortage quantity - Average leftover quantity
Ordering quantity = 124.68 + 10 - 0
Ordering quantity = 134.68 pounds (approx.)
Therefore, Cynthia should order approximately 134.68 pounds of oysters each day to meet the expected demand and minimize the risk of stockouts while considering the cost of excess inventory.
To determine how many pounds Cynthia should order each day, we need to calculate the expected daily demand and then factor in the costs and revenues associated with ordering different quantities of oysters.
Step 1: Calculate the expected daily demand:
Given:
Mean (µ) = 100 pounds
Standard deviation (σ) = 15 pounds
The formula for the expected daily demand from a normal distribution is:
Expected Daily Demand = Mean
Therefore, the expected daily demand is 100 pounds.
Step 2: Calculate the expected daily profit for each order quantity:
For each order quantity, we need to calculate the profit by considering the costs and revenues associated with that quantity.
Let's assume Cynthia orders x pounds of oysters each day.
Cost of purchasing x pounds of oysters = $5 * x
Revenue from selling x pounds of oysters = $9 * x
Revenue from selling unsold oysters to her cousin = $2 * (100 - x)
Total profit = Revenue - Cost
Total profit = ($9 * x + $2 * (100 - x)) - ($5 * x)
Total profit = $9x + $200 - $2x - $5x
Total profit = $2x + $200
Step 3: Determine the optimal order quantity:
The optimal order quantity is the one that maximizes the total profit. To find this, we need to find the value of x that yields the maximum total profit.
To determine the optimal order quantity, we can use the derivative of the total profit equation with respect to x, set it equal to zero, and solve for x.
d(Total profit)/dx = 0
2 - 2x = 0
2x = 2
x = 1
Therefore, the optimal order quantity for Cynthia should be 1 pound of oysters each day to maximize her profit.
Note: It's worth mentioning that optimizing the order quantity based on the expected daily demand does not take into account other factors like storage capacity, shelf life of oysters, or customer satisfaction. These factors should also be considered when determining the order quantity.