Problem $4$: $($Land$$s End$)$ Geoff Gullo owns a small firm that manufactures $$Gullo Sunglasses.$$ He has the

opportunity to sell a particular seasonal model to Land$$s End. Geoff offers Land$$s End two purchasing

options:

$$ Contract #$1.$ Geoff offers a price of $$65$ and agrees to credit Land$$s End $$53$ for each unit Land$$s End

returns to Geoff at the end of the season. Since styles change each year, there is essentially no value in the

returned merchandise for Geoff Sunglasses.

$$ Contract #$2.$ Geoff offers a price of $$55$ for each unit, but returns are NOT accepted. In this case, Land$$s

End throws out unsold units at the end of the season $($no salvage value for Land$$s End$).$

This season$$s demand for Land$$s End can be assumed to follow a normal distribution with a mean of $200$ and a

standard deviation of $125.$ Land$$s End sells those sunglasses for $$100$ each. Geoff Gullo$$s production cost is

$$25$ each.

Part I: Land$$s End:

a$)$ For Land$$s End, what is the overage and underage cost of its ordering decision under contract is #$1?$

What is the optimal order quantity for Land$$s End under this contract?

b$)$ For Land$$s End, what is the overage and underage cost of its ordering decision under contract is #$2?$

What is the optimal order quantity for Land$$s End under this contract?

To identify the better option for Land$$s End, let$$s compare the expected profits for the two contracts.

c$)$ What are the expected lost sales quantities for the two contracts respectively? $($Lost sales $=$ standard

deviation of demand $\backslash$times $(),$ where $()$ can be found in the standard normal table.$)$