Lily Lawson wants to run a lemonade stand during the summer holidays. She has picked a spot near a park and expects to capture daily joggers, casual strollers and families spending their day out as potential customers. To cater to such a diverse crowd, she plans to sell $$lemonade by millilitres$$ instead of forcing them to buy a fixed volume. She plans to sell at $$5$ per litre $(200$ ml at $$1$ or $450$ ml at $$2\mathrm{.}25,$ etc.$).$

Since making lemonade at the stand requires an outdoor production license, she decides to make the batch at home before she goes to the park. Based on the cost of lemons, sugar and other overheads, she estimates it would cost her $$1\mathrm{.}50$ per litre. Of course, she can$$t go back home and create a new batch during the day, and any leftover lemonade at the end of the day needs to go down the drain.

She surveys a few friends and family members to arrive at an estimated daily demand normally distributed with a mean of $12\mathrm{.}5$ litres and a standard deviation of $1\mathrm{.}5$ litres.

$2\mathrm{.}1.$ How many litres of lemonade should she produce each day? Please provide your

answer rounded to the nearest millilitre. $[1]$

$2\mathrm{.}2.$ What is the expected profit based on this quantity? Please provide your answer

rounded to the nearest dollar.$[1]$

Note: Please evaluate around $1000$ demand scenarios to get a reliable outcome.

$2\mathrm{.}3.$ Please comment on the distribution of the profit figure around this expected value.

You can plot a histogram using a sample of demand values to get an insight. $[1]$

$2\mathrm{.}4.$ Lily has noticed that customers who approach the stand after she runs out of lemonade are less likely to visit her in future. She estimates the opportunity cost of such lost sales as $$2$ per litre. Should this insight drive her to produce more or less

lemonade? Justify your answer. $[1]$

$2\mathrm{.}5.$ After talking to some lemonade experts, Lily has found a way to salvage the unsold

lemonade. If she adds a unique herb to the lemonade, the leftover lemonade can be refrigerated and reused the next day. Adding this herb costs her $$0\mathrm{.}10$ per litre, but she doesn$$t want to change the price charged to the customer. Overnight refrigeration costs her $$1$ per litre. Based on this information, what should be the $$salvage value$$ of the unsold stock? $[1]$

$2\mathrm{.}6.$ Using the stockout penalty in $(4)$ and salvage value in $(5),$ what should Lily$$s revised lemonade production estimate be$?$ Please provide your answer rounded to the nearest millilitre. $[1]$