$1)$ An office routinely replenishes their inventory of printer paper. The usage rate is $55$ packs per week, and there is no significant seasonality. The fixed cost of a replenishment is estimated to be $$34,$ and an annual interest of $0\mathrm{.}35$ is used by the company. The supplier of the paper regularly offers the following all$-$units discount structure:

For $0<$ Q $<450$ units, unit cost $=$ $$12$

For $450<$ Q units, unit cost $=$ $$10\mathrm{.}5$

a$.(12$ points$)$ What replenishment size should be used?

b$.(4$ points$)$ The supplier is interested in having office acquire at least $650$ units at a time. What is the largest unit price they could charge to encourage the office to purchase $650$ units?

$2)$ A company that produces medical devices and they order many of their parts using a four$-$month time supply. They$$ve just been following past practices and want you to re$-$analyze their inventory management practices. An analyst estimated the values below:

D $=4750$ units$/$year

A $=$ $$5\mathrm{.}00$ per order

v $=$ $$2\mathrm{.}15$ per units

r $=0\mathrm{.}25$ $$/$$$/$year

a$.(4$ points$)$ What is the economic order quantity of the item?

b$.(3$ points$)$ What is the time between consecutive replenishments of the item when the EOQ is used?

c$.(3$ points$)$ Assuming that the lead time is $3$ days, at what inventory level should an order be placed? $($Assume $365$ days per year$)$

d$.(6$ points$)$ The production manager insists on using his simple four$-$month supply rule and he says it$$s the cheaper option. Determine if the answer from part a or the four$-$month supply rule is better.

$3)$ A restaurant makes its own bread for the lunch and dinner service every day. On average, they use $2450$ pounds of flour per day. The restaurant manager places an order with a local supplier and it is delivered at a rate of $4000$ pounds per day after a lead time of $6$ days. The cost of placing, receiving and handling an order for bulk flour delivery is $$150$ per order placement. The cost per pound is $$1\mathrm{.}25.$ The interest rate they use is $0\mathrm{.}25$ $$/$$$/$year$.$

a$.(6$ points$)$ Determine the optimal lot size in bags. Let one bag be $50$ pounds.

b$.(8$ points$)$ Determine the $($i$)$ maximum inventory, $($ii$)$ average inventory, $($iii$)$ time between orders, and $($iv$)$ reorder point. Note: Use pounds of flour

c$.(3$ points$)$ What is the inventory cost for the year?

$4)$ A hotel regularly purchases new bedding to replace old or spoiled bedding throughout the year. Their supplier just announced a special one$-$time discount of products they are no longer going to produce because they will release a new line soon. This supplier charges an ordering cost of $$5\mathrm{.}00,$ interest of $0\mathrm{.}05$ $$/$$$/$month$,$ and $$40$ per bedding set. The hotel usually uses about $25$ bedding sets per month. The discount is $35\%$ off the original price.

a$.(5$ points$)$ What is the optimal order quantity, assuming they currently have $10$ bedding sets in inventory

b$.(6$ points$)$ How much money will the hotel save?