Finding the optimum supply quantity when there is uncertainty in forecast This is actually a quiz that will be needed in a real life food stall! I need to decide how much stock to supply for my pumpkin soup stall. I sell each soup for $5$ dollars a cup, and let's say my ingredients cost is $1$ dollar. Therefore an outcome of under-forecasting is $4$ dollars per unit, while an outcome of over-forecasting is 1 dollar per unit. 
My forecast isn't so simple, however. I'm guessing that the most probable number of sales is $150$ units, but I'm very unsure, so there's a normal distribution behind this prediction with a standard deviation of $30$ units. 
This is harder than I expected.
Intuitively I would prepare ingredients for $180$ units, at which point I'd guess that the likely opportunity costs that would come with understocking would roughly meet the likely costs of overstocking. But given this is such a common dilemma, I thought that someone must be able to find a precise solution, and would then hopefully be able to explain it in layman's terms. 

If you get supplies for $n$ bowls and there are $m$ customers, your profit is $p=\begin {cases} 4n \text{ if } n \le m \\ 5m-n \text{ if } n \gt m \end {cases}$
The expectation of the profit is then $\sum pP(m)$ where $P(m)$ is the probability distribution of the number of customers.  Unfortunately, the normal distribution is not normal-real life distributions usually have much longer tails.  In this case, you probably have greater than one chance in 3 million of being too sick that day and not selling anything.
That aside, your intuition that you should err on the side of having too much is correct.  If you take the expected profit of preparing $n+1$ bowls and subtract the expected profit of preparing $n$ bowls, the difference is $p(n+1)-p(n)=4\sum_{m \gt n}P(m)-\sum_{m\le n}P(m)$.  If you know (or have a guess at) $P(m)$ you can goal seek this in Excel or use your favorite 1D rootfinder to solve this.
