May 02, 2023

### Question 1. 30 Points, Each Part Is Worth 5 Points.

We have the following information about daily demand for one-pint bottles of organic milk at a convenience store:

 Average daily demand: 40 bottles Standard deviation of daily demand: 8 bottles Order lead time: 3 days Standard deviation of order lead time: 1.5 days Item cost: \$1.8 per bottle Cost to place an order: \$5 per order Yearly holding cost per bottle: 30% of item cost

<LL><ITEM><P><INST>a. </INST>What is the economic order quantity? What is the resulting total annual relevant cost?

EOQ=Sqrt ( 2* Annual Demand * Order cost/ Holding Cost) = sqrt(40*365*2*5/0.54)= 520 approx

Holding Cost = 520*1.8*.3/2= \$ 140.4

Number of orders = 365*40/520*5=\$ 140.4

Product cost = 40*365*1.8=\$ 26280

Total Relevant Cost = 26280+ 140.4+ 140.4=\$ 26560.8

</P></ITEM><ITEM><P><INST></P></ITEM>

<ITEM><P><INST>bbbb. </INST>Suppose the store is currently ordering six cases of 100 bottles at a time (i.e., the order quantity is 600). How does the annual relevant cost of this policy compare with the cost you obtained in part (a)? Show your work.

Holding Cost = 600*1.8*.3/2= \$ 162

Number of orders = 365*40/600*5=\$ 122

Product cost = 40*365*1.8=\$ 26280

Total Relevant Cost = 26280+ 162+ 122=\$ 26563.7

The cost under the current policy is higher than the EOQ cost by \$ 3 approx and thus cost is minimum at EOQ. Further, there is no large variation in cost.

c. Suppose the store places an order every time the inventory level drops to 200. What is the probability of no stock-out achieved under this policy?

Probability of No stock out =

Demand during lead time = 40*3* Service Level * 1.5= 120

SD of Daily Demand =8

SD of Lead time demand = 8* Sqrt (3* Service Level * 1.5)* Service Level= 13.86

Value of Service Level= Probability

85.03% Probability that there shall be no stock out.

</P></ITEM>

<ITEM><P><INST>d. Now assume that the store wants to achieve a probability of no stock-out of 99% (z=2.33). </INST>What is the new reorder point? How much of the reorder point consists of safety stock?

Reorder days = 3+ 2.33*1.5= 6.49 Days

Mean Demand during reorder period= 40*6.49= 259.8

Safety Stock= 8*Sqrt(6.49)* 2.33= 47.5

Total Reorder level = 259.8+ 47.5= 307.30

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