Acme Clothing Company is anticipating the upcoming winter season and needs to place an order for boots with its supplier, Boots Unlimited. Acme knows that it can’t sell pairs of boots for more than $54 and that excess inventory can be dumped at a 50% discount. Boots Unlimited has quoted Acme a $40 price for each pair of boots. Acme has forecasted demand for its pairs of boots at 400, with a normally distributed standard deviation of 300 pairs of boots.

What should Acme's order quantity be to maximize its profit, given these constraints?

Acme Clothing Company should order approximately 415 pairs of boots for the upcoming winter season by using the newsvendor model and critical ratio calculations to balance costs and demand uncertainty.

Acme Clothing Company's Boot Order Decision

Acme Clothing Company needs to decide how many pairs of boots to order from Boots Unlimited for the upcoming winter season.

The key factors to consider are:

Sale price per pair of boots: $54
Excess inventory discount: 50%
Quoted price per boot from supplier: $40
Forecasted demand: 400 pairs with a standard deviation of 300 pairs

To determine the optimal order quantity, Acme Clothing should use a technique like the newsvendor model, which is ideal for scenarios with uncertain demand. By balancing the cost of overstock (selling at a 50% discount) and understock (losing potential sales), Acme can estimate the best quantity of boots to order.

Given the quoted purchase price of $40 and the maximum sale price of $54, the potential loss from excess inventory is significant. Acme should calculate the critical ratio to find the optimal order quantity. The critical ratio (CR) is given by CR = (Selling Price - Cost Price) / (Selling Price - Discounted Price).

Using this formula:

Selling Price = $54
Cost Price = $40
Discounted Price = $27 (50% of $54)
CR = ($54 - $40) / ($54 - $27) = $14 / $27 = 0.518

Next, Acme needs to use the critical ratio to find the corresponding z-score from the standard normal distribution, which is approximately 0.05 (use z-tables or normal distribution calculators for precise values). Then, Acme can calculate the optimal order quantity by applying the z-score to the forecasted demand distribution:

Optimal Order Quantity = Mean Demand + (Z-score * Standard Deviation) = 400 + (0.05 * 300) = 415 pairs

Therefore, Acme Clothing should order approximately 415 pairs of boots to optimize its inventory for the winter season.