# Optimization problem

A friend of mine sells $k$ models of blenders. Some of the blenders are very simple and cheap, others are very sophisticated and more expensive. His data consists, for each month, of the prices of each blender (which are fixed by him), and the number of sold units for each model. To establish some notation, he knows for months $j=1,\dots,n$ the vectors $$(p_{1j},\dots,p_{kj}) \qquad \textrm{and} \qquad (n_{1j},\dots,n_{kj}) \, ,$$ where $p_{ij}$ is the price of blender model $i$ during month $j$, and $n_{ij}$ is the number of sold units of blender model $i$ during month $j$.

Given the data, he wants to determine prices $(p^*_1,\dots,p^*_k)$ which maximize the value of his expected future sales.

I have some ideas on how to start modeling this problem with some sort of Poisson regression, but I really don't want to reinvent the wheel. It would also be nice to prove that the desired maximum exists under certain conditions. Would someone please give me pointers to the literature of this kind of problem?

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I would really like to hear the rationale behind the downvote on this question! The only possibility I can imagine at present is that there was some concern about what the statistical nature of this question might be. However, it seems clear to me that there is a statistical component given that the sales data can be viewed as random counts from some underlying distribution. I suppose an edit that draws this point out a little more clearly might help. But, my comments here are quite speculative. (+1) – cardinal Sep 16 '12 at 19:17
Tks, cardinal. I will edit it in the next few days and add info that will clarify the inference aspects of the solution. – Zen Sep 17 '12 at 1:08

Suppose there is a function $f(\cdot)$ which takes the prices, $\vec{p}$, of all $k$ blenders and returns the number of sales, $\vec{n}$. Then, the problem is:

$$\underset{\vec{p}}{\arg\max}\;\; \vec{p}^T f(\vec{p})$$

The solution to this problem will depend on the assumptions you want to make. I would go with the simplest model that comes to my mind, first. Let's assume that the number of sales of a blender depends on its own price only and not on others' prices. That is, the number of sales of each blender is independent. This assumption allows us to break the vector valued function $f(\cdot)$ into $k$ scalar functions. We have $f_i:p \mapsto n,\;\;i=1,\dots,12$, and the problem becomes:

$$\underset{\vec{p}}{\arg\max} \sum_{i=1}^k p_i f_i(p_i)$$

Now we have to assume a model for $f_i(\cdot)$. We can again try a simple (linear) form: $f_i(p) = \alpha_i p + \beta_i$. For each blender, you can estimate the parameters ($\alpha_i, \beta_i$) of this function using the historical sales data. Once, they are estimated, optimizing the cost function above should be straightforward and will give you the optimal prices you are looking for.

As you mentioned in your post, you can assume a Poisson model for $f(\cdot)$, too.

That the sales of blenders are independent from each other is probably a naive assumption (because customers will look at many blenders, compare them and then buy one). So, I would go for the vector valued $f(\cdot)$ and start with linear modeling. The optimization shouldn't be too difficult.

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