Question: John Smith owns a small firm that manufactures "Smith Sunglasses." He has the opportunity to sell a particular seasonal model to Target. John offers Target two purchasing options:

Option 1: John offers to set his price at $65 and agrees to credit Target $50 for each unit Target returns to John at the end of the season (because those units did not sell). Since styles change each year, there is essentially no value in the returned merchandise.

Option 2: John offers a price of $53 for each unit, but returns are no longer accepted. In this case, Target throws out unsold units at the end of the season.

Assume (for simplicity) that this season’s demand for this model follows the following probability distribution. Target will sell those sunglasses for $100 each. John’s production cost is $25.

Probability Distribution of Demand for Smith Sunglasses

1. How much would Target buy if they choose option 1?
   (the term \(-15\left( C-X \right)\) is because we can return the items, but we lose $15 for each returned item. The following table shows a table with the values for C and the respective expected profit:

   C	Expected Value
   80	2800
   100	3400
   120	3900
   140	4250
   160	4400
   180	4350
   200	4150

   
   (See the attached spreadsheet to see the formula that computes the expected value)
   The optimal purchase is 160 pair of glasses, for a profit of $4,400.
2. How much would Target buy if they choose option 2?
   (See the attached spreadsheet to see the formula that computes the expected value)
   The optimal purchase is 140 pair of glasses, which a maximum expected profit of $5,280.
3. Which option will Target choose? Why?
4. Suppose Target chooses option 1 and orders 140 units. What is John Smith’s expected profit?

Price: $2.99

Solution: The downloadable solution consists of 3 pages
Deliverable: Word Document