I am tackling a linear maximization problem where I need to select the optimal product among several options over a series of weeks, given certain constraints, in order to maximize future profit. The decision variables are binary, indicating whether or not a particular product is chosen.

The objective function is expressed as:

$$ \max_{j} \sum_{i} choice(j) \cdot c_{i,j}^T p_{i,j}$$


$c_{i,j}$ represents the production quantity of product $j$ in week $i$, $p_{i,j}$ represents the price of product $j$ in week $i$, $choice(j)$ is a binary variable indicating the selection of product $j$, and The summation is over all weeks $i$.

However, I encounter a challenge because $p_{i,j}$ is not fully known. It is a time series, and I lack future values as well as extensive historical data. To address this issue, I'm considering either Robust Optimization or Stochastic Optimization techniques. Which approach should I pursue, and how can I implement it effectively to maximize profit in this scenario?

  • $\begingroup$ it's not a real answer but my advice is to estimate the $p_{ij}$ and use the estimates. How you optimize ( robust, stochastic whatever ) isn't going to matter if the $p_{ij}$ are way off. So, the safest thing to do is get good estimates. How you do that I can't say but its wasn't clear what was missing and why. ? Also, what data is available for estimation ? $\endgroup$
    – mark leeds
    Apr 3 at 4:20
  • $\begingroup$ The problem is i don't have much history of $p_{ij}$, only 3 years, optimization doesn't take much time only 1s, so doing 100-1000 optimization for different values of $p_{ij}$ is feasible. $\endgroup$
    – anasse
    Apr 4 at 14:55
  • $\begingroup$ Hi Anasse: It's definitely feasible but if you don't know what $p_{ij}$ are, how does an optimization help ? 3 years of data seems like a good amount to me but I'm unfamiiar with the data and what time interval etc. Basically, you need to get good estimates of $p_{ij}$. $\endgroup$
    – mark leeds
    Apr 5 at 4:35
  • $\begingroup$ the data i have is from 2021, price per week of each product, so say i have 159 entries, and i want to maximize projected profit over the next year, so basically i need to make sure that the optimal chosen product isn't too sensitive to infered prices. $\endgroup$
    – anasse
    Apr 5 at 16:05
  • $\begingroup$ right. if you're interested in an approximate solution, then I think you're approach is correct. Run many scenarios with different values for $p_{ij}$ and see how the result changes depending on the chosen value of $p_{ij}$. But there must be ways to estimate the $p_{ij}$ with limited data. I don't know what they are because I don't understand what data is available. $\endgroup$
    – mark leeds
    Apr 6 at 17:42


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