If The Kingdom and Alexandria are acting independently, what is their expected understock quantity if they order the optimal quantity?
Understand the Problem
The question presents a scenario where Gregory and his followers are selling body armor. The demand for the body armor is normally distributed with a mean of 250 units and standard deviation of 50. The question asks us to calculate the expected understock quantity if The Kingdom and Alexandria order the optimal quantity, so we need to determine how many units are likely to be less than the optimal quantity.
Answer
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Answer for screen readers
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Steps to Solve
- Calculate the critical ratio
The critical ratio is calculated as $ CR = \frac{Cu}{Cu + Co} $ where $Cu$ is the cost of underage and $Co$ is the cost of overage The cost of underage is the profit made per unit which is $65 - $3 = $62 The cost of overage is the cost of the armor which is $3 $ CR = \frac{62}{62 + 3} = \frac{62}{65} = 0.9538 $
- Find the Z-score
Find the Z-score corresponding to the critical ratio using a standard normal distribution table. A critical ratio of 0.9538 gives a Z-score of approximately 1.68.
- Calculate optimal order quantity
Optimal order quantity = Mean + (Z-score * Standard Deviation) Optimal order quantity = $ 250 + (1.68 * 50) = 250 + 84 = 334 $
- Calculate the expected understock
The expected understock can be computed using the formula $ Standard,Deviation * E(z) $, where $ E(z) $ is the expected value for the loss function for the value z (1.68 calculated in a previous stem) Using a Standard Normal Loss Function table, for z = 1.68, $ E(z) = 0.0155 $ Expected Understock = $ 50 * 0.0155 = 0.775 $
- Round to the nearest whole number.
Since we're dealing with units, we round the expected understock to the nearest whole number. Expected Understock ≈ 1
- Account for the fact that two parties (The Kingdom and Alexandria) are ordering
Since The Kingdom and Alexandria act independently, we multiply the expected understock quantity by 2 Total Expected Understock Quantity ≈ 1 * 2 = 2
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More Information
The expected understock quantity essentially represents the average number of units by which demand will exceed the order quantity.
Tips
A common mistake involves forgetting to account for both The Kingdom and Alexandria acting independently. Another mistake could be using Safety Stock formula for newsvendor problems.
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