What is the optimal quantity of boxes for the professor to bring back home to sell to his friends? The souvenirs cost the professor $125 a box and he sells them for $290 a box. Sou... What is the optimal quantity of boxes for the professor to bring back home to sell to his friends? The souvenirs cost the professor $125 a box and he sells them for $290 a box. Souvenirs that dry out due to age can be sold for $80. The demand for boxes has a mean of 80 with a standard deviation of 20. Read more

Steps to Solve

1. Determine the Cost of Overstocking (Co) The cost of overstocking is the cost of buying a box of souvenirs minus the salvage value if it doesn't sell at full price.

   $Co = \text{Cost per box} - \text{Salvage value per box}$ $Co = $125 - $80 = $45$

2. Determine the Cost of Understocking (Cu) The cost of understocking is the profit lost for each box that could have been sold at full price but wasn't because of insufficient inventory.

   $Cu = \text{Selling price per box} - \text{Cost per box}$ $Cu = $290 - $125 = $165$

3. Calculate the Service Level (SL) The service level is the probability that demand will be met. It's calculated as:

   $SL = \frac{Cu}{Cu + Co}$ $SL = \frac{165}{165 + 45} = \frac{165}{210} = 0.7857$

4. Find the Z-score Corresponding to the Service Level We need to find the z-score that corresponds to a cumulative probability of 0.7857. We can use a standard normal distribution table or a calculator to find this value. The z-score is approximately 0.79.

5. Calculate the Optimal Inventory Level Now we calculate the optimal inventory level using the following formula:

   $\text{Optimal Inventory} = \text{Mean Demand} + Z \times \text{Standard Deviation}$ $\text{Optimal Inventory} = 80 + 0.79 \times 20 = 80 + 15.8 = 95.8$

6. Round to the Nearest Whole Number Since we can't have a fraction of a box, we round the optimal inventory level to the nearest whole number.

   $\text{Optimal Inventory} \approx 96$