One of the products sold by Office Min at its online store is the HP Deskjet XYZ$23$ printer. We know the following about this product:

Average monthly demand $($assume $52$ weeks per year$)$: $80$ printers

Standard deviation of weekly demand: $10$ printers

Order lead time: $2$ weeks

Standard deviation of order lead time: $0\mathrm{.}2$ weeks

Item cost: $$150$ per printer

Cost to place an order: $$30$ per order

Yearly holding cost per monitor: $30\%$ of item cost

Desired service level during reordering period: $99\%($z $=2\mathrm{.}33)$

a$.$ What is the economic order quantity for this printer?

b$.$ Calculate annual ordering costs and holding costs $($ignoring safety stock$)$ for the EOQ. What do you notice about the two?

c$.$ Suppose Office Min currently orders $200$ printers at a time. How much more or less would Office Min pay in holding and ordering costs per year if it ordered $40$ printers at a time? Show your work.

d$.$ What is the reorder point for the printer? How much of the reorder point consists of safety stock?

e$.$ Because electronics becomes obsolete so quickly, Office Min is thinking about raising holding cost from $40\%$ of item cost to a higher percentage. What will be the impact on the economic order quantity? Briefly explain your answer.

For parts f and g$,$ use the following formula to consider the impact of safety stock $($SS$)$ on average inventory levels and annual holding costs:

f$.$ What is the annual cost of holding inventory, including the safety stock? How much of this cost is due to the safety stock?

g$.$ Suppose Office Min is able to cut the lead time to a constant one week. What would the new safety stock level be$?$ How much would this reduce annual holding costs?

Question $2.12$ points

You and a couple of friends from the MBA program decide to sell T$-$shirts during the Spring MBA Commencement to commemorate the event. The T$-$shirts sell for $$15$ and cost $$5$ to make plus $$1$ to transport to the commencement site. After the commencement weekend, you think you can continue to sell them online but only at $$5$ per T$-$shirt. In addition, for each shirt you sell online, you will incur a shipping cost of $$2.$ Suppose the demand for shirts can be approximated with a normal distribution and the mean demand is $100$ shirts with a standard deviation of $25.$ How many shirts should you produce?

Question $3.26$ points

Consider a manufacturer of personal computers that sells PCs through the $4$ Best Buy stores in the Triangle area. The weekly demand at the Durham store is normally distributed with mean $30$ and standard deviation $10.$ The weekly demand at each of the Cary and two Raleigh stores is normally distributed with mean $25$ and standard deviation $6.$ Inventory is reviewed on a continuous basis and the lead time from the manufacturer$$s plant to the retail stores is $8$ days.

a$.$ Please compute the safety stock levels at each of the four retail stores if the goal is to achieve a probability of no stock$-$out of $98\%.(10$ points$)$

b$.$ If each computer costs $$1200$ and the holding cost rate is $30\%$ per year per dollar, what is the total annual cost of carrying safety inventory across the four retailers? $(6$ points$)$

c$.$ The PC manufacturer decides that they don$$t need the retailers to sell their computers. They decide to withdraw their inventory from the retail stores and satisfy customer demand from a central warehouse in Cary. Assume that customers are indifferent between buying at a retail store versus waiting for their computer to be shipped from the warehouse, i$.$e$.$ demand does not change. The warehouse will also follow a continuous inventory review policy and the lead time from the plant is $6$ days. Please compute the annual cost of carrying safety inventory under this supply chain design using the cost parameters from part $($b$).(10$ points$)$

Part a $(16$ points, $10+6)$

Consider a company producing and selling single use coffee pods. The amount of coffee in a pod is one of the quality metrics they measure. When the process was known to be under control, the company collected seven samples with four measurements in each. The numbers in the table represent grams of coffee per pod. Below is that data:

Sample $1234567$

Measurement $111\mathrm{.}8210\mathrm{.}8511\mathrm{.}07411\mathrm{.}5748\mathrm{.}72610\mathrm{.}978\mathrm{.}926$

Measurement $28\mathrm{.}92611\mathrm{.}0710\mathrm{.}38\mathrm{.}9268\mathrm{.}98610\mathrm{.}110\mathrm{.}05$

Measurement $310\mathrm{.}3410\mathrm{.}08311\mathrm{.}0210\mathrm{.}1911\mathrm{.}0748\mathrm{.}938\mathrm{.}926$

Measurement $48\mathrm{.}92611\mathrm{.}1810\mathrm{.}458\mathrm{.}92611\mathrm{.}2311\mathrm{.}1510$

i$.$ Please calculate the centerlines, the upper and lower control limits for the Xbar and R charts the company can use to monitor if the process is remaining in control.

ii$.$ A few weeks later, the quality manager collected the following three samples. Is the process still in control? Please answer using the control charts you developed for part $($i$).$

Part b $(11$ points, $6+5)$

A third$-$party packaging company has a process capable of filling coffee cups with $11\mathrm{.}5$ grams of coffee and standard deviation of $0\mathrm{.}21$ grams. The company in question $4$ part $($a$)$ specifies the upper and lower tolerance