Question

samantha brons sells fresh salmon at her store daily. she estimated that the demand for salmon follows a normal distribution with a mean of 118 pounds and a standard deviation of 10 pounds. she pays $8 for a pound of fish, whichk sells for $26. any fish not sold that day are sold to another store for $2 per pound. according to this given information, how many pounds should she order everyday?

Answer 1

Samantha should order approximately 1 pound of salmon daily.

Step-by-step explanation:

Step 1: Calculate the daily demand

To determine how many pounds Samantha should order daily, we need to calculate the daily demand. Since the demand for salmon follows a normal distribution, we can use the mean and standard deviation provided. We know that the mean demand is 118 pounds and the standard deviation is 10 pounds. To find the daily demand, we need to find the z-score for the desired inventory level.

Z-score = (Inventory Level - Mean Demand) / Standard Deviation = (Inventory Level - 118) / 10

Let's assume Samantha wants to have an inventory level that ensures a 90% chance of meeting demand. This corresponds to a z-score of 1.645.

Therefore, 1.645 = (Inventory Level - 118) / 10. Solve for the inventory level:

1.645 * 10 + 118 = Inventory Level

Inventory Level = 16.45 + 118 = 134.45

Step 2: Calculate the order quantity

Since Samantha pays $8 per pound of fish and sells each pound for $26, her profit per pound is $26 - $8 = $18. If she doesn't sell the fish, she can sell it to another store for $2 per pound, resulting in a loss of $8 - $2 = $6 per pound. Based on this, we can calculate the expected profit per pound:

Expected Profit = (Probability of Selling * Profit per Pound Selling) + (Probability of Not Selling * Profit per Pound Not Selling)

The probability of selling is the area under the normal distribution curve to the right of the z-score. Using a standard normal table, we can find this probability to be approximately 0.9474. The probability of not selling is 1 - 0.9474 = 0.0526.

Expected Profit = (0.9474 * $18) + (0.0526 * -$6)

Expected Profit = $17.0548 - $0.3156 = $16.7392

To calculate the order quantity, we divide the expected profit by the profit per pound:

Order Quantity = Expected Profit / Profit per Pound = $16.7392 / $16

Order Quantity = 1.0462 pounds

Step 3: Round the order quantity

Since Samantha cannot order fractional pounds, she should round the order quantity to the nearest whole pound. In this case, she should order approximately 1 pound of salmon daily.