Problem Statement:

Patty King needs to decide how many patties to prepare each morning. The patties are freshly made and cannot be stored for more than a day due to health and quality standards.

Cost of Making a Patty: $$1\mathrm{.}50$ each

Selling Price of a Patty: $$3\mathrm{.}00$ each

Unsold Patties' Disposal Cost: $$0\mathrm{.}50$ each $($the store needs to pay for the cost of waste management$)$

Demand Scenario:

The daily demand for patties follows a normal distribution.

From historical sales data, the average daily demand is $200$ patties, with a standard deviation of $50$ patties.

Calculate the optimal newsvendor quantity of patties that Patty King should prepare to maximize its expected profit.