# Robust or Stochastic Optimization Approach for Maximizing Profit with Limited Price Information

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:

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

Where:

$$c_{i,j}$$ 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?

• 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 ? Apr 3 at 4:20
• 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. Apr 4 at 14:55
• 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 $p_{ij}$. Apr 5 at 4:35
• 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. Apr 5 at 16:05
• 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. Apr 6 at 17:42