Organic Soap Co$.(15$ points$)$

Sarah$$s Organic Soap Company makes four kinds of organic liquid soap$-$regular$,$lavender$,$citrus$$ and $$tea tree$.$ Demand for the four scents are $150,120,75$ and $50$ kgs per hour respectively. Sarah$$s production process can produce any soap at the rate of $450$ kgs per hour but $1\mathrm{.}5$ hours are needed to switch between scents. During those switchover times, the process doesn$$t produce any soap. Sarah wants to choose a production schedule that $($i$)$ cycles repeatedly through the four scents, $($ii$)$ meets the required demand and $($iii$)$ minimizes the amount of inventory held.

How many kgs of $$regular$$ should Sarah produce before switching over to another scent?

Sarah needs to purchase organic Palm oil to make her soaps. She needs $1,000$ kgs of Palm oil per day on average. The supplier delivers immediately and charges a $$60$ delivery fee per order $($which is independent of the order size$)$ and $$4\mathrm{.}75$ per kg$.$ Sarah$$s annual holding cost is $25\%.$ Assume $52$ weeks per year and $5$ days per week. If Sarah wants to minimize inventory holding and ordering costs, how much Palm oil should she purchase with each order $($in kgs$)?$

c$.$ If Sarah purchased the EOQ per order, what would be her average inventory holding and delivery fees per day $($in $s$)?($Note$,$ do NOT include her purchasing costs per day.$)$

Sarah$$s supplier is willing to sell her Palm oil at a $5\%$ discount if she purchases $10,000$ kgs at a time. If she were to purchase $10000$ kgs per order, what would be her average inventory holding and delivery fees per day $($in $s$)?($Note$,$ do NOT include her purchasing costs per day.$)$

Should Sarah order $10,000$ units at the $5\%$ discount or order the original EOQ $($i$.$e$.$ without the discount$)?($Hint: Compare the total costs in each case, that is$,$ the sum of purchase, deliver, and average holding costs.$)$

How often will the manufacturer receive an order? $($Round to the nearest integer.$)$

$[$Note: Less$-$than$-$truckload shipment refers to the case where the shipment quantity is less than the capacity of a standardized truck.$]$

Because of the manufacturer$$s current ordering needs, components are delivered in LTL $($less$-$than$-$truckload$)$ shipments. Manufacturer A is contemplating initiating a pool distribution agreement with another manufacturer $($called $$manufacturer B$)$ located in the same industrial park. In other words, these manufacturers agree to order and manage inventory together to meet their combined demand. Manufacturer B buys components from the same supplier and its plant is also open $365$ days per year. However, manufacturer B uses these components at an average rate of $10$ components per day. It incurs the same holding cost as manufacturer A and because it places orders with the same supplier, it also incurs $$240$ to place an order.

What is the EOQ for the pool distribution agreement? If they split the fixed ordering cost and the quantity received in each shipment in proportion to their demands, how much will manufacturer A save on average in inventory and ordering costs?

Should Sarah order $10,000$ units at the $5\%$ discount or order the original EOQ $($i$.$e$.$ without the discount$)?($Hint: Compare the total costs in each case, that is$,$ the sum of purchase, deliver, and average holding costs.$)$