Please Make sure to answer question $2.$

EOQ and Power$-$of$-$Two reorder intervals: Powered by Koffee $($PBK$)$ is a new campus coffee store. PBK uses $55$ bags of whole bean Arabica coffee every month, and demand is steady throughout the year. PBK has signed a contract to purchase its coffee from a special supplier, Phish Roasters, for a price of $ $25$ per bag and a fixed cost of $$100$ per order $($for every delivery, independent of the order size$).$ PK incurs an annual inventory cost rate of $25\%$ per year, applied to the unit cost; further, there is an additional storage cost of $$1$ per bag per month. Assume that each month is equivalent to four working weeks.

a$)(10$ points$)$ EOQ for Arabica: Determine the optimal amount of Arabica coffee PBK should order from its supplier in each order, and the corresponding reorder interval $($i$.$e$.,$ the time between orders$)$ in weeks. Calculate the annual holding and setup cost incurred by your EOQ ordering policy.

b$)(15$ points$)$ Power$-$of$-$two Ordering for Arabica: Calculate a good reorder interval, the corresponding order quantity, and the annual holding and setup cost, if PBK wants the reorder interval to be a power$-$of$-$two in weeks, e$.$g$.,1,2,4,8,16,32...$ weeks. Why is your reorder interval good? Why might a power$-$of$-$two reorder interval be used in practice instead of the Economic Order Interval that minimizes relevant costs?

c$)(*25$ 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.

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

ii$)(*10$ 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.

iii$)(*10$ 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.

$2)(25$ points$)$ Newsvendor: Consider a lesser$-$known variety of coffee beans called Liberica with a unique woody flavor that some love and others hate. Liberica coffee beans are offered by a different $($Asian$)$ supplier Insomniac Roasters, who operate on a periodic delivery schedule, so you must order once a month. There is no fixed order cost. PBK plans to sell Liberica coffee and the demand forecast for an upcoming month is normally distributed with a mean of $40$ bags and a std$.$dev. of $10$ bags. If you over order $($too much$),$ it costs you $15$$ per bag leftover in a month. If you under order $($too little$),$ it costs you $25$$ per bag.

a$)(10$ points$)$ Newsvendor Quantity, Probability: How many bags of Liberica coffee would you order from Insomniac Roasters for the upcoming month? For your order quantity, what would be the chance of having $10$ or more bags of Liberica coffee leftover at the end of that month?

b$)(*15$ points$)$ Simulation: Suppose you order $50$ bags of Liberica coffee for the upcoming month. Consider simulating the monthly demand with realized demand values of $25,35,45,55,$ and $65$ bags. For each realized demand quantity: i$)$ List the sales quantity. ii$)$ Calculate the corresponding realized total underage $+$ overage costs incurred. iii$)$ For which realized demand values out of the sample of $25,35,45,55,$ and $65,$ will the realized underage $+$ overall cost be $>200$$$?$ Answers may be presented as a table with rows for realized demand values and columns for sales quantity, total cost, Yes $/$ No for total cost $>200$$$.$