$(*25$ points$)$ Two$-$product System: Suppose PBK can also buy whole bean Robusta coffee from its supplier, Phish Roasters. Robusta coffee has twice the caffeine of Arabica, a stronger flavor, but Robusta coffee beans have lower taste quality than Arabica. For simplicity, assume that sale of Robusta coffee does not affect the sale of Arabica coffee. PBK will use $25$ bags of Robusta coffee every month. Phish Roasters$$ price for Robusta coffee is $$35$ per bag, with a fixed cost of $$100$ per order. However, if PK orders both Robusta and Arabica coffee concurrently, the total fixed cost per order is only $$125$ per order and not $$100+100$ per order, i$.$e$.,$ joint ordering saves on the total fixed order cost per order. PK uses an annual inventory cost rate of $25\%$ per year for each type of coffee with a $$1/$ bag, month extra storage cost.

$(5$ points$)$ Calculate a power$-$of$-$two reorder interval for Robusta coffee.

$(*10$ points$)$ Would you advise PK to order Arabica and Robusta coffee beans independently using their respective EOQ or would you advise PK to coordinate orders for the two types of coffee beans? Show annual cost calculations to support your recommendation and, if you coordinate briefly explain how the coordination would work.

$(*10$ points$)$ In what way does a supply replenishment lead time $($assumed to be a constant number of weeks$)$ affect the EOQ model? What happens to the optimal order quantity $($EOQ value$)$ and reorder interval $($T$*)$ for a type of coffee $($e$.$g$.,$ Robusta$)$ when its replenishment lead time is equal to two weeks instead of being instantaneous $($as sometimes assumed in EOQ$)?$ When $($at what time epochs$)$ will replenishment orders be placed? Calculate the value of the reorder point, defined as the on$-$hand inventory level at which a replenishment order is triggered.