BrightSide Produce is a student$-$driven, nonprofit social venture designed to eliminate food insecurity in underserved neighborhoods. Food insecurity can result from lack of access, availability, or means to purchase healthy foods. In San Diego County in $2022,$ approximately $323,000$ adults and $163,000$ children were experiencing food insecurity, with many more considered to be at$-$risk. BrightSide provides a service that increases the accessibility, availability, and affordability of fresh fruits and vegetables by serving as a produce distributor that reaches food insecure customers in low$-$income communities.

These neighborhoods have, at most, a low number of supermarkets and a disproportionate number of food stores that primarily stock unhealthy foods and beverages. The US Department of Agriculture refers to these areas as food deserts. Small food stores are limited in their ability to carry healthy products, including fresh produce, often because they cannot meet minimum order requirements $($e$.$g$.,$ a food store may be required to purchase at least $$250$ in produce for a distributor to service the store$).$ Based out of San Diego State University $($SDSU$),$ BrightSide is a full$-$service produce distributor that is able to serve these stores by "breaking bulk" and giving stores the flexibility to buy the varieties and quantities that are appropriate for the stores at low prices, without minimum order requirements. ${?}^1$

BrightSide buys a wide variety of fresh fruits and vegetables from local farms and a local distributor. They then sell and deliver these products weekly to store owners. They also provide shelves and baskets at no cost to the store. One such store is T$\text{'}$s Liquor Store $($disguised name$)$ in National City, as seen in the photos below.

Dr$.$ Iana Castro is the Co$-$Founder and Executive Director of BrightSide Produce, and a professor of Marketing at SDSU, Aware of the challenge of inventory management at BrightSide, she welcomed the help of Gabrifel Garzo, a graduating MBA student from USD. Gabriel had taken an MBA elective where he learned many inventory management tools that he thought could apply to BrightSide. Specifically, because the produce is perishable, he decided to apply the newsyendor model. He gathered data from the past year for each store and each item that BrightSide sells. He began his analysis with just one item, avocados, before expanding the solution to the dozens of other items that BrightSide offers. Here is the summary data $($total across all stores$),$ for avocados:

Average weekly demand $=220$ units

Standard deviation of weekly demand $=39\mathrm{.}56$ units

Cost per unit for BrightSide $=$$$0\mathrm{.}86$

Average sales price to the stores per unit $=$$$1\mathrm{.}03$

Gabriel struggled with the salvage value associated with any items that go bad on store shelves. In order to provide incentives for the stores to sell this healthy produce, BrightSide buys back any obsolete products at the exact price the store paid$-$ in this case $$0\mathrm{.}86$ per avocado. During its weekly store visits, BrightSide removes all items that have gone bad and then donates or composts them. Gabriel realized therefore that the overage cost for BrightSide was $$0\mathrm{.}86$ per unit, or equivalently, the salvage value was $$0.$

With this information, please answer the following questions.

a$)$ What is BrightSide's optimal order quantity for avocados?

b$)$ If they decide on this order quantity, what would be the in$-$stock probability?

c$)$ When Gabriel saw these numbers, it seemed that they could work against the goal of "availability" at the stores. With this in mind, if BrightSide decided to override the optimal order quantity and use $240$ units instead, what would be the new in$-$stock probability?