O'Neill Inc. is a designer and manufacturer of fashionable water sports apparel and equipment. To produce some of their items, they work with TEC. Consider a specific item for which O'Neill's seasonal demand forecast is a Normal distribution with a mean of $27396$ and a standard deviation of $6250$ TEC charges $$152\mathrm{.}5$ per unit $($including transportation$).$ O'Neill sells each unit for $245$ during the season and can sell the leftovers at the end of the season for $$47\mathrm{.}5$ per unit.

Assuming a single order for the season, how many of these items should O'Neill order? Calculate the resulting seasonal expected profits for both O'Neill and TEC,'assuming TEC's unit cost is $60\%$ of its selling price for the units produced for this pre$-$season order.

Suppose, observing significant expected lost sales, O'Neill would like to improve their ability to meet demand, and they bring a proposal to TEC. Accordingly, if O'Neill needs any extra units during the season, TEC will quickly deliver them and collect a $20\%$ higher unit price than the pre$-$season order. Assume that TEC's unit cost for in$-$season production orders is $75\%$ higher than pre$-$season orders.

Assuming that TEC agrees with this offer and produces only what $O^{\text{'}}N$ Neill orders, calculate the resulting seasonal expected profits for both O'Neill and TEC.

Suppose TEC makes a counteroffer with a pair of contract alternatives as described below.

Buyback Contract: For any leftovers, TEC would reimburse O'Neill up to $50\%$ of the purchase price. If O'Neill wishes to buy any extra units during the season, TEC will deliver them at $$200$ per unit.

Options Contract: O'Neil would purchase options at $$35$ and exercise at $$122\mathrm{.}5$ per unit. If O'Neill wishes to buy any extra units beyond the options created, TEC will deliver them at $$180$ each. TEC can salvage any of its leftovers at $$45$ per unit on average.

Consider the buyback contract above. What are the resulting expected profits for O'Neill and TEC, assuming TEC produces only what O'Neill orders?

Consider the options contract above. What are the resulting expected profits for O'Neill and TEC, assuming TEC produces only what O'Neill orders?

Consider the options contract above. Suppose TEC can freely decide how many options to produce, and O'Neill ends up accepting TEC's decision. How many units $($options$)$ would TEC produce? What are the resulting expected profits for O'Neill and TEC?

To maximize the total supply chain profit, how many units should TEC produce pre$-$season and in$-$season? What is the resulting expected total supply chain profit?

Design a revenue$-$sharing contract $($find its parameters$)$ that maximizes O'Neill's profits while giving TEC the maximum of their expected profits you find in questions $1,2\mathrm{.}3,$ and $4$

If you had to include holding costs in your analysis, how would the order quantities found in the first $5$ questions change? For each question $(1$ through $5),$ comment "increase" or "decrease" and very briefly state why.