$($Inventory Management $-$ Part $1)$ FCC Cafeteria, which is a fast lunch spot in $$ankaya University campus, purchases several basic cooking ingredients from a supplier. The annual usage $($demand$),$ purchasing cost, and setup $($order$)$ cost of these ingredients are given in the table below. The inventory carrying rate is $0\mathrm{.}10$ per unit per year.

$\backslash$table$[[$Ingredient$,$A$,$B$,$C$],[$Annual Demand $($in units$),1,000,2,000,1,000],[$Setup Cost $($$ $/$ order$),2,4,3],[$Purchasing Cost $($$ $/$ unit$),4,1,6]]$

$($a$)$ Assume that all ingredients are ordered independently. What is the total inventory carrying cost of all ingredients when all ingredients are optimally ordered?

$($b$)$ The cafeteria management wants to order all ingredients in each cycle, which has a fixed common length for all ingredients. What is the optimal length of this cycle time?

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$($c$)$ Assume that all ingredients are jointly ordered from a supplier. The cafeteria management wants to order all ingredients in each cycle, which has a fixed common length for all ingredients. The major setup $($order$)$ cost associated with a group of ingredients is $$141$ per order, and the minor ordering cost of each ingredient is as given in the table above. Calculate this common cycle time and the total annual setup $($order$)$ cost when all ingredients are optimally ordered?

$($d$)$ Assume that all ingredients are jointly ordered from a supplier, and all ingredients have varying order cycles. The major setup $($order$)$ cost associated with a group of ingredients is $$141$ per order, and the minor ordering cost of each ingredient is as given in the table above. What is the ordering frequency for the ingredient $C($i$.$e$.,m_C)$ if you use the heuristic algorithm discussed in the lecture?