Optimizing profit in a practical setting, best technique?

[The Investor]

You are selling apples from a stand at the market.
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?

 

[Stephen Tashi]

To get a mathematical answer, you'll have to make specific assumptions to turn most of the things that are unknown in this problem into things that are "given". I'm sure you can get advice about what assumptions to make. Unless you make them you are going nowhere mathematically.

My advice is to use simulation. Creating a simulation is a good mental discipline It forces you to fill-in enough "givens" to have a model of the situation. it also gives you a tool to test the effect of various strategies.

 

[The Investor]

Hi Stephen. Thanks for the reply.

I was hoping to solve (or at least arrive at a decent practical approximation to a solution) this mathematically by using information as it becomes available.

So say you have a) 0.5, b) 200 and c) 20%. You change this to a) 0.48 for a time and then to 0.52. Would recursive Bayesian estimation be appropriate to the problem for example?

As I understand it, this can be seen as multiparameter optimization problem?

 

[Chestermiller]

Basically, you are doing a set of experiments that have built-in random variations. You first want to map out your function space (profit vs input variables), and you want to use a functionality that can capture the non-linear interaction between the inputs. Look up Design of Experiments on Google. This discipline has developed strategies for doing what you want to be able to do using the minimum number of experiments. It typically involves varying the inputs in combinations, rather than one at a time.

Chet

 

[Stephen Tashi]

As I understand it, this can be seen as multiparameter optimization problem?

Your problem combines two mathematical problems that are rarely tackled simultaneously. These are:

1) Given a set of data, find a function that models how that data is generated. (A "function" may or may not be a simple formula - it could be a complicated computer simulation.)


2) Given a known function of several variables, find the values of the variables that maximize or minimize the function. ( "Parameters" can be regarded variables when they are unknown),


If you had solved 1) then solving 2) to find the optimal values for a,b,c would be a "multivariable" optimization problem.


To solve 1), you must make assumptions.

For example, the "design of experiments" approach mentioned by Chestermiller assumes the function is multinomial function of the variables with unknown coefficients. You plan tests and then use the results to estimate the coefficients from the data.

If you want enter the realm of Bayesian estimation, you still need to solve 1). Instead of fitting a multinomial function to the data, you would be fitting a stochastic process to the data. Again, I recommend that you do that by writing a simulation (or several different simulations). That will force you to supply enough given information to define a specific mathematical problem.

Perhaps you are seeking a method that solves 1) and 2) simultaneously, updating the estimated solution as more data comes in. This would still involve making enough assumptions to define a specific mathematical problem. Numerical data, by itself, does not supply enough "given" information to define or solve problems involving probability. "Mathematics" does not have any methods that solve probability problems when given only the bare facts of some numerical data.

 

[The Investor]

Thanks guys, I'll look into experimental design.

 

[wigglywoogly]

Am I understanding the broad scheme of your question:

You want to develop a function for profit (or expected profit), which takes those 3 variables a, b and c as arguments? Then you want to take the total derivative of this function (to account for feedback loops), and set this equal to zero to find a critical point (maximum)?