You are selling apples from a stand at the market.(adsbygoogle = window.adsbygoogle || []).push({});

You are making a profit to comfortably cover your costs and provide compensation for effort.

Now you are looking to increase profitability.

The aim is to optimize the following variables to achieve maximum (expected) profit:

a) Price for an individual apple

b) Apples in stock each day

c) % Discount rate for bulk purchase (5 or more apples)

We assume that the effects of changes in the variables are straightforward, for a) a decrease in price will lead to more sales (expected) and an increase will lead to fewer sales(expected), there is assumed to be a single optimal price, although the (type of) distribution around this is unknown. For b) Having more apples in stock will allow us to sell more apples, but also risk having left over apples that will spoil, having fewer apples minimises risk of being left with unsold goods that will spoil, but it also means we may run out of apples and make less than we could have done. For c) The discount rate for bulk purchases allows us to sell more, but we also may lose income from people who would have been willing to pay the full unit price for bulk purchases.

The variables are correlated (bring the price down drastically without increasing apples in stock and we are far more likely to run out etc.). We have no information on exactly how they are correlated beyond what 'real world' common sense would suggest. We have to build up an estimate of correlations as we go along.

At the beginning of each day, you are allowed to change a, b and c as you wish. You then have to stick to that until the next day, when you may change them again. For the sake of simplicity assume that optimal values of these variables are the only thing of relevance to success. It's given that any step closer to the optimal value for each variable will give a higher expected profit, although you need to take into account that if lowering prices leads to running out of stock, you need to increase stock and vice versa.

There is a degree of randomness in the results. So if from one day to the next you lower the price of an apple, it's possible that you will sell fewer apples than the previous day. The degree of randomness is unknown, and has to be estimated as you gather more data day by day.

So the question is, what kind of strategy/technique/algorithm would be most appropriate to solving the above, to achieve maximum (expected) profit?

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# Optimizing profit in a practical setting, best technique?

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