Multi$-$periodic inventory $(7$ marks$)$

A retailer operates $365$ days of the year and sells one product that it purchases from a

supplier. The annual demand of the product is $90,000$ units with an equal daily demand across

the year with the standard deviation of $48$ units. The supplier offers the retailer with the

following pricing depending on the order quantity:

$$ For any order quantity below $5,000$ units, the supplier offers the base rate for product$$s

price for $$150$ per unit and the base rate of the ordering cost $($including delivery$)$ for

$$1,000$ per order

$$ For any order quantity between $5,000$ and $9,999$ units, the retailer will receive a $1\%$

discount on the base rate of the products$$ price and a $5\%$ discount on the base rate of the

ordering cost.

$$ For any order quantity of $10,000$ units or more, the retailer will receive a $2\%$ discount on

the base rate of the product$$s price and a $10\%$ discount on the base rate of the ordering

cost$.$

Based on the negotiation, the supplier agrees that it will take six days to fulfil $($deliver$)$ the

retailer$$s order. The products are kept at the retailer$$s storage area with a maximum capacity

of $8,000$ units, and the product$$s annual holding cost per unit is $6\%$ of the product$$s unit price.

Based on the information above, answer the following questions:

a$.$ Calculate the minimum annual total inventory costs for each order quantity range, and

determine the order quantity and the number of orders that produces the lowest total

inventory cost on each range $(2$ marks$)$

b$.$ What order quantity would you recommend to the retailer? Explain your reasons $(1$ mark$)$

c$.$ If the retailer only wants to carry safety stock of $100$ units, what will be the service level

and the re$-$order point of the products? $(1$ mark$)$

d$.$ If the retailer wants to set the service level to $85\%,$ suggest two possible options that it can

take to achieve the goal without changing the demand. Show the calculations to support

your suggestions as well as factors that need to be considered in the suggestions. $(2$ marks$)$

e$.$ If the retailer decides to change the ordering system into a fixed$-$time period model with a

review period of $10$ days and a service level of $90\%,$ what will be the safety stock and the

order quantity of the new ordering system based on the original information? $(1$ mark$)$