$1)$ Harold is considering starting a sandwich$-$making business to earn some extra income. Harold divides his needs into three areas: breads, meats and cheeses, and condiments. He estimates that he will be able to use all the products he purchases before they spoil, so perishability is not a relevant issue. The demand and cost parameters are as follows.

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Bread

Meat and Cheese

Condiments

Weekly Demand

$12$ Packages

$6$ Packages

$2$ pounds

Cost per unit

$$1\mathrm{.}25$

$$3\mathrm{.}50$

$$1\mathrm{.}50$

Additional order cost

$$6\mathrm{.}00$

$$2\mathrm{.}00$

$$4\mathrm{.}00$

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The fixed order cost is $$6\mathrm{.}00$ every time an order is placed. Assume that holding costs are based on an annual interest rate of $25$ percent and that a year has $52$ weeks.

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a$)$ Find the optimal quantities that Harold should purchase of each type of product when they are ordered independently and calculate the total annual order and holding.

b$)$ Find the optimal quantities that Harold should purchase of each type of product when all products are ordered at same time and calculate the total annual order and holding.

c$)$ Find the optimal quantities that Harold should purchase of each type of product when products are ordered jointly but not every order contains every product and calculate the total annual order and holding.

d$)$ Compare the total annual costs from a$,$ b$,$ and c$.$ Which policy is better.

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