High School

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Newsvendor Problem:

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, which sells for $26. Any fish not sold that day are sold to another store for $2 per pound.

According to this information, how many pounds should she order every day?

(Do not round your intermediate calculations. Round your final answer to the nearest whole number. Only enter the number; do not enter any units.)

Answer :

Final answer:

The newsvendor problem for Samantha requires finding the critical ratio and corresponding z-score to calculate the optimal order quantity of salmon. Information provided about demand, costs, and prices is used in formulas to derive the quantity that balances the risk of overstocking against the risk of stockouts.

Explanation:

To solve Samantha Bron's newsvendor problem, we need to use the newsvendor model, which is a fundamental model in operations management for determining the optimal inventory level. Samantha needs to balance the cost of ordering too much (which leads to excess inventory that is sold at a lower price) against the cost of ordering too little (which leads to missed sales). The problem provides the following information:

  • Demand for salmon follows a normal distribution.
  • The mean (μ) of the demand is 118 pounds.
  • The standard deviation (σ) of the demand is 10 pounds.
  • Cost per pound of fish (c) is $8.
  • Selling price per pound (r) is $26.
  • Sale price of unsold fish (s) is $2.

We use these values to find the critical ratio, which is given by:

(r - c) / (r - s) = ($26 - $8) / ($26 - $2)

The critical ratio is then transformed into a z-score that indicates how many standard deviations above the mean Samantha should set her inventory level to meet the demand at this critical ratio.

After calculating the critical ratio, we can use z-score tables or a statistical software to find the corresponding z-value. Finally, we calculate the optimal order quantity Q* using the formula:

Q* = μ + zσ

Once we find the z-value, we can calculate how many pounds of salmon she should order daily.

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