Part I:

Background

ITA Kitchen Equipment $($ITA$)$ manufactures cordless mixers for use in the kitchens of consumers. ITA sells to retailers, which sell the mixers to the ultimate consumer. One of ITA's retail customers is Almost Best Buy $($ABB$).$ On January $1,$ ITA sells to and receives payment from ABB for $100$ cordless mixers with a one$-$year warranty for $$50$ each. The mixers are delivered by ITA to ABB upon receipt of payment and the warranty is initiated at that time as well by ITA. This warranty provides for a replacement of the mixer if the mixer fails to work properly within one year of the date of purchase. ITA also sells its mixers with no warranty for $$40$ per unit. The cost to manufacture each mixer is $$32.$

ITA also provides its retail customers with sales incentives in the form of volume discounts on purchases of mixers with warranties paid at the end of an annual period. The agreement between ITA and ABB provides for the following volume discounts. Additionally, the probability of purchases for each volume level as estimated by ITA is provided based on historical experience and forecasted sales.

$\backslash$table$[[$Number of mixers purchased,Discount,Probability$],[$Less than $1,000,0\mathrm{.}00\%,35\mathrm{.}00\%$