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I have the following problem:

My business consists of $n$ different departments $D_{1,...,n|Y}$. I want to decide how much money $A_{i|Y}$ I should allocate to each department $D_{i|Y}$ to maximize profit $P_Y$, given that I have decided what my overall budget $B_Y$ for the year is fixed.

I also have historical data of budgets, allocations for each department and profits from past years $1,...,Y-2,Y-1$:

$A_{1|1},...A_{n|1}$, $B_1$, and $P_1$

$A_{1|2},...A_{n|2}$, $B_2$, and $P_2$

....

$A_{1|Y-1},...A_{n|Y-1}$, $B_{Y-1}$, and $P_{Y-1}$

My questions: How to solve this problem? Is this a regression problem, an optimization problem, or some combination of both? and how to account for the times series element from the fact that I am using historical actuals?

Off the top of my head, this looks like an optimization problem:

Given $B_Y$, find the optimal set of allocations $A_{1|Y},...A_{n|Y}$ that maximizes $P_Y$, with the constraint that $\sum{A_{i|Y}}=B_Y$.

Except that I don't have any way of calculating $P_Y = f(A_{1|Y},...A_{n|Y})$.

So finding $f$ is a regression/supervised learning problem.

I'm thinking I might be able to use GLM to find $P_Y = f(A_{1|Y},...A_{n|Y})$ from the historical data, then use Linear Programming to find the optimal $A_{1|Y},...A_{n|Y}$ that maximizes $P_Y$.

But I have some doubts about this approach:

  1. Is it safe to assume that the relationship between profit and allocations is linear ? How would I know if I can use a linear model?
  2. If assume that a linear model doesn't work, and that I have a to use some non linear or non parametric model, then what method can I use to perform the optimization part? Is gradient descent applicable in this case? What other methods are possible?
  3. Even if I've answered 1) and 2), I'm still not accounting for the time series factor involved - that I don't a have static regression problem, but a set of values co-varying over time and that time might be a factor in determining $P_Y = f(A_{1|Y},...A_{n|Y})$. How do I account for the effect of time in my model?

Additional thought: How is this related to portfolio optimization in Finance, in the sense that do they share the same mathematical structure?

How would one account for risk in my problem?

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This is more a comment than a solution because, disclaimer, I don't know what the correct solution is.

The two main approaches I would recommend trying are L2-penalisation LASSO and dynamic linear models.

L2-penalisation LASSO model in theory should find the optimal allocation of budget within a limit. In particular I think L2 would be better suited over L1 as L1 tends to result in more 0 coefficient parameters which I am assuming would not be an appropriate solution. The downside here is that it doesn't account for any time dependencies or non-linear relationships.

Its been a long time since I studied dynamic linear models but my memory is telling me that they can essentially be thought of as a time varying regression model. This would help account for the time series nature but again doesn't help with the budget limits or non-linear relationships.

Personally I would run both of the above and see if they provide similar solutions. If they do then great thats good evidence for a solution, if they don't then indeed there is a more complicated underlying process for which I personally don't know how to account for :)

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First of all you want to combine regression and arima to create a transfer function for each profit center as regression by itself is woefully inadequate to deal with temporal data. Then with a reasonable set of prediction equations that have incorporated latent variables proxied by pulses/level shifts/local time trends ( robust transfer function) you can set out an objective function for optimization given constraints.

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