Assignment $4(50$ points$)$

$1.$ Consider a retailer that sells kids$$ toys. The retailer is planning its inventory for the festive season. In particular, the retailer wants to decide how many units of LEGO$$s $($LEGO is a toy manufacturer$)$ latest playset to procure. This playset that the retailer is considering, was developed by LEGO especially for the festive season. Since LEGO sells its products through multiple retailers and also through its own website, the supply of this playset is tight; it seems that the retailer will have to order the entire stock that it intends to procure for the festive season, in one go$.$

The retail price of the playset that is charged to customers who buy it$,$ is $$400.$ The retailer contacts LEGO, and the two parties agree on a contract. To procure each unit from LEGO, the retailer has agreed to pay $$300.$ At the end of festive season, the retailer does not see much scope for selling the playset to customers. To reduce such concerns, LEGO agrees to buy the unsold units of the playset back from the retailer at the end of the festive season at $$250$ for each unit, and LEGO thinks that it can sell these at a discount through its own website later. The retailer has also conducted some market research to forecast the demand for the particular playset, and it sees the demand to be Normally distributed with an average of $10000,$ and a standard deviation of $500.(5$ points for each part in $($a$)-($i$).$ Show all work for all parts$)$

$($a$)$ What is the Marginal Profit and Marginal Loss for the retailer in this contract? Which of the two is greater?

$($b$)$ Calculate the probability of not selling the entire inventory $($or the probability of having excess inventory$)$ when the optimal quantity is ordered, using the formula discussed in class.

$($c$)$ What is the optimal quantity of playsets that the retailer should procure? Is the optimal quantity less than, equal to$,$ or greater than the average demand?

$($d$)$ Fast$-$forward to the festive season of the following year. The retailer wants to procure units of another new playset from LEGO. This time, LEGO has learned that it is difficult to sell the units that are bought back from retailers after the festive season, and is not willing to pay a lot to the retailer to buyback the unsold units. Consequently, the LEGO agrees to buyback the unsold units at $$200$ per unit. All other terms of the contract, retail price, and the demand distribution remain the same. What is the marginal profit for the retailer now? What is the marginal loss? Which of the two is greater?

$($e$)$ For the new contract mentioned in $($d$),$ what is the optimal quantity of playsets that the retailer should procure? Is the optimal quantity less than, equal to$,$ or greater than the average demand?

$($f$)$ Fast$-$forward another year, the retailer and LEGO are negotiating a contract again for the festive season. This time, LEGO is willing to pay even lower amount to buyback the unsold units from retailer. The two parties agree that LEGO will now pay $$150$ for each unit of unsold inventory that the retailer has. All other terms of the contract, retail price, and the demand distribution remain the same. What is the marginal profit for the retailer now? What is the marginal loss? Which of the two is greater?

$($g$)$ For the new contract mentioned in $($f$),$ what is the optimal quantity of playsets that the retailer should procure? Is the optimal quantity less than, equal to$,$ or greater than the average demand?

$($h$)$ LEGO has been reducing the price at which it buys unsold units back from the retailer over the years. What is the effect of this on the retailer$$s optimal order quantity? Is it decreasing, increasing, or unaffected?

$($i$)$ How do the comparisons between the marginal profit and marginal loss that you made in parts $($a$),($d$),$ and $($f$),$ affect whether the retailer$$s corresponding order quantity is greater$/$less than the average demand?

$($j$)$ As LEGO goes on decreasing the price at which it buys unsold units back from the retailer, for which of the two parties $($retailer or LEGO$)$ does the risk of loss from unsold inventory increase? For which of the two parties $($retailer or LEGO$)$ does the risk of loss from unsold inventory decrease?