You have rented a location and are almost ready to launch your new restaurant in Brooklyn. The main

things you still need to do is to choose prices and to hire the service staff.

You can choose any price point for the dishes between $20 and $50 (in increments of $1). For simplicity, we will assume all dishes in your new restaurant will have the same price. The cost of goods sold is $10 per dish served. Higher prices will lead to more revenue per dish, but they will also decrease demand. You have forecasted that your expected demand per night is:

Expected Demand = 120 - 2*Price.

That is, if you charge $20, you expect to serve 80 dishes a night, but if you charge $50, you only expect to serve 20 dishes a night. Actual demand is probabilistic and you believe a Poisson distribution is appropriate to capture variability.

At the same time, you must decide how many servers to hire, which can be any number between 1 and 10. Each server costs $180 a night in salary and benefits. A server can serve 12 customers in a night with the quality level you desire. If you have more customers than 12*Servers in a given night, you will call that a bad service night. You'd like to keep the number of bad service nights to under 10% of nights.

What's the optimal price point and staffing level for optimizing profits while still ensuring the probability of a bad service night is under 10%?